Content-Type: multipart/mixed; boundary="-------------0403120808222" This is a multi-part message in MIME format. ---------------0403120808222 Content-Type: text/plain; name="04-80.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-80.keywords" Schroedinger operator, magentic field, Aharonov Bohm effect, delta interaction, Landau level, singular perturbation, canonical commutation relation, self-adjoint extension ---------------0403120808222 Content-Type: application/postscript; name="Aharonov_Bohm_Solenoids.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Aharonov_Bohm_Solenoids.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.78 p1.4 Copyright 1996-98 ASCII Corp.(www-ptex@ascii.co.jp) %%dvipsk 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: Aharonov_Bohm_Solenoids.dvi %%Pages: 60 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: /usr/local/bin/dvips-ASCII -f -N0 %+ 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a(as) p 1130 2941 a(it) p 1241 2941 a(has) p 1429 2941 a(an) p 1578 2941 a(imp) s(ortance) p 2104 2941 a(in) p 2231 2941 a(gauge) p 2521 2941 a(theories.) p 2964 2941 a(In) p 3099 2941 a(the) p 3281 2941 a(presen) m(t) p 3634 2941 a(pap) s(er,) 0 3062 y(w) m(e) p 148 3062 a(consider) p 532 3062 a(the) p 705 3062 a(sp) s(ectral) p 1075 3062 a(problem) p 1459 3062 a(for) p 1612 3062 a(the) p 1785 3062 a(Sc) m(hr\177) p 1971 3062 a(odinger) p 2323 3062 a(op) s(erators) p 2758 3062 a(in) p 2877 3062 a(a) p 2962 3062 a(plane) p 3227 3062 a(with) p 3453 3062 a(a) p 3539 3062 a(non-zero) 0 3182 y(uniform) p 375 3182 a(magnetic) p 801 3182 a(\014eld) p 1022 3182 a(in) p 1145 3182 a(addition) p 1539 3182 a(to) p Fn 1667 3182 a(\016) p Fo 1714 3182 a(-lik) m(e) p 1935 3182 a(magnetic) p 2361 3182 a(\014elds.) p 2676 3182 a(The) p 2886 3182 a(op) s(erator) p 3288 3182 a(of) p 3408 3182 a(this) p 3608 3182 a(t) m(yp) s(e) p 3836 3182 a(is) 0 3302 y(studied) p 342 3302 a(b) m(y) p 477 3302 a(Nam) m(bu) p 819 3302 a(\([Nam]\)) p 1185 3302 a(and) p 1374 3302 a(Exner,) 1690 3277 y(\024) 1687 3302 y(St'o) m(v) p 1892 3302 a(\023) p 1903 3302 a(\020) m(\024) p 1931 3302 a(cek) p 2102 3302 a(and) p 2292 3302 a(Vyt) p 2449 3302 a(\024) p 2455 3302 a(ras) p 2612 3302 a(\([Ex-St-Vy]\).) 146 3423 y(Let) p Fn 321 3423 a(N) p Fo 438 3423 a(=) p 542 3423 a(1) p Fn(;) p Fo 635 3423 a(2) p Fn(;) p Fo 728 3423 a(3) p Fn(;) p 821 3423 a(:) p 865 3423 a(:) p 909 3423 a(:) p Fo 966 3423 a(or) p Fn 1086 3423 a(N) p Fo 1202 3423 a(=) p Fk 1306 3423 a(1) p Fo(.) p 1477 3423 a(Let) p Fk 1652 3423 a(f) p Fn(z) p Fm 1747 3438 a(j) p Fk 1784 3423 a(g) p Fm 1834 3386 a(N) 1834 3447 y(j) p Fl 1867 3447 a(=1) p Fo 1993 3423 a(b) s(e) p 2126 3423 a(p) s(oin) m(ts) p 2419 3423 a(in) p Fj 2533 3423 a(R) p Fl 2621 3381 a(2) p Fo 2693 3423 a(and) p 2883 3423 a(put) p Fn 3062 3423 a(S) p Fm 3122 3438 a(N) p Fo 3217 3423 a(=) p Fk 3321 3423 a([) p Fm 3387 3386 a(N) 3387 3447 y(j) p Fl 3420 3447 a(=1) p Fk 3514 3423 a(f) p Fn(z) p Fm 3609 3438 a(j) p Fk 3646 3423 a(g) p Fo(.) p 3767 3423 a(W) p 3859 3423 a(e) 0 3543 y(assume) p 337 3543 a(that) p Fn 1500 3663 a(R) p Fo 1602 3663 a(=) p 1710 3663 a(inf) p Fm 1706 3723 a(j) p Fg 1739 3723 a(6) p Fl(=) p Fm(k) p Fk 1848 3663 a(j) p Fn(z) p Fm 1921 3678 a(j) p Fk 1980 3663 a(\000) p Fn 2080 3663 a(z) p Fm 2125 3678 a(k) p Fk 2168 3663 a(j) p Fn 2223 3663 a(>) p Fo 2326 3663 a(0) p Fn(:) p Fo 3701 3663 a(\(1.1\)) 0 3868 y(This) p 223 3868 a(assumption) p 739 3868 a(is) p 837 3868 a(satis\014ed) p 1212 3868 a(if) p Fn 1301 3868 a(N) p Fo 1422 3868 a(is) p 1520 3868 a(\014nite.) p 1808 3868 a(De\014ne) p 2110 3868 a(a) p 2191 3868 a(di\013eren) m (tial) p 2679 3868 a(op) s(erator) p Fk 3072 3868 a(L) p Fm 3141 3883 a(N) p Fo 3240 3868 a(on) p Fj 3376 3868 a(R) p Fl 3463 3826 a(2) p Fk 3524 3868 a(n) p Fn 3596 3868 a(S) p Fm 3656 3883 a(N) p Fo 3756 3868 a(b) m(y) p Fk 1337 4077 a(L) p Fm 1406 4092 a(N) p Fo 1501 4077 a(=) p Fq 1604 4077 a(p) p Fl 1666 4036 a(2) p Fm 1666 4101 a(N) p Fn 1734 4077 a(;) p Fq 1875 4077 a(p) p Fm 1937 4092 a(N) p Fo 2032 4077 a(=) 2146 4009 y(1) p 2146 4053 49 4 v Fn 2154 4145 a(i) p Fk 2204 4077 a(r) p Fo 2310 4077 a(+) p Fj 2408 4077 a(a) p Fm 2471 4092 a(N) p Fn 2538 4077 a(;) p Fo 0 4283 a(where) p Fn 294 4283 a(i) p Fo 377 4283 a(=) p Fk 502 4205 a(p) p 585 4205 127 4 v 585 4283 a(\000) p Fo(1) p 757 4283 a(and) p Fk 959 4283 a(r) p Fo 1091 4283 a(=) p 1216 4283 a(\() p Fn(@) p Fm 1305 4298 a(x) p Fn 1350 4283 a(;) p 1394 4283 a(@) p Fm 1445 4298 a(y) p Fo 1486 4283 a(\)) p 1569 4283 a(is) p 1680 4283 a(the) p 1861 4283 a(gradien) m(t) p 2256 4283 a(v) m(ector) p 2561 4283 a(with) p 2796 4283 a(resp) s(ect) p 3142 4283 a(to) p 3274 4283 a(the) p 3455 4283 a(co) s(ordinate) p Fn 0 4403 a(z) p Fo 77 4403 a(=) p 181 4403 a(\() p Fn(x;) p 318 4403 a(y) p Fo 370 4403 a(\)) p Fk 435 4403 a(2) p Fj 529 4403 a(R) p Fl 616 4361 a(2) p Fo 656 4403 a(.) p 725 4403 a(W) p 817 4403 a(e) p 890 4403 a(assume) p 1225 4403 a(that) p 1433 4403 a(the) p 1598 4403 a(magnetic) p 2013 4403 a(v) m(ector) p 2303 4403 a(p) s(oten) m(tial) p Fj 2712 4403 a(a) p Fm 2775 4418 a(N) p Fo 2870 4403 a(=) p 2973 4403 a(\() p Fn(a) p Fm 3062 4418 a(N) s(;x) p Fn 3185 4403 a(;) p 3229 4403 a(a) p Fm 3280 4418 a(N) s(;y) p Fo 3400 4403 a(\)) p 3468 4403 a(b) s(elongs) p 3815 4403 a(to) p Fn 0 4524 a(C) p Fg 77 4488 a(1) p Fo 152 4524 a(\() p Fj(R) p Fl 277 4482 a(2) p Fk 339 4524 a(n) p Fn 412 4524 a(S) p Fm 472 4539 a(N) p Fo 539 4524 a(;) p Fj 583 4524 a(R) p Fl 671 4482 a(2) p Fo 710 4524 a(\)) p Fk 771 4524 a(\\) p Fn 860 4524 a(L) p Fl 926 4488 a(1) p Fm 926 4548 a(l) q(oc) p Fo 1017 4524 a(\() p Fj(R) p Fl 1143 4482 a(2) p Fo 1182 4524 a(;) p Fj 1226 4524 a(R) p Fl 1313 4482 a(2) p Fo 1353 4524 a(\).) p 1465 4524 a(The) p 1667 4524 a(function) p 2050 4524 a(rot) p Fj 2191 4524 a(a) p Fm 2254 4539 a(N) p Fo 2322 4524 a(\() p Fn(z) p Fo 2409 4524 a(\)) p 2477 4524 a(=) p 2583 4524 a(\() p Fn(@) p Fm 2672 4539 a(x) p Fn 2716 4524 a(a) p Fm 2767 4539 a(N) s(;y) p Fk 2911 4524 a(\000) p Fn 3011 4524 a(@) p Fm 3062 4539 a(y) p Fn 3104 4524 a(a) p Fm 3155 4539 a(N) s(;x) p Fo 3278 4524 a(\)\() p Fn(z) p Fo 3403 4524 a(\)) p 3475 4524 a(represen) m(ts) 0 4644 y(the) p 168 4644 a(in) m(tensit) m(y) p 567 4644 a(of) p 678 4644 a(the) p 846 4644 a(magnetic) p 1263 4644 a(\014eld) p 1475 4644 a(p) s(erp) s(endicular) p 2092 4644 a(to) p 2212 4644 a(the) p 2380 4644 a(plane.) p 2678 4644 a(W) p 2770 4644 a(e) p 2846 4644 a(assume) p 3183 4644 a(that) 1226 4877 y(rot) p Fj 1368 4877 a(a) p Fm 1431 4892 a(N) p Fo 1498 4877 a(\() p Fn(z) p Fo 1585 4877 a(\)) p 1651 4877 a(=) p Fn 1755 4877 a(B) p Fo 1856 4877 a(+) p Fm 1984 4770 a(N) p Fi 1955 4794 a(X) p Fm 1954 4977 a(j) p Fl 1987 4977 a(=1) p Fo 2093 4877 a(2) p Fn(\031) t(\013) p Fm 2263 4892 a(j) p Fn 2299 4877 a(\016) p Fo 2346 4877 a(\() p Fn(z) p Fk 2456 4877 a(\000) p Fn 2556 4877 a(z) p Fm 2601 4892 a(j) p Fo 2638 4877 a(\)) p 3701 4877 a(\(1.2\)) 1927 5214 y(1) p eop %%Page: 2 2 2 1 bop Fo 0 631 a(in) p Fk 109 631 a(D) p Fg 189 595 a(0) p Fo 212 631 a(\() p Fj(R) p Fl 337 589 a(2) p Fo 377 631 a(\)) p 443 631 a(\(the) p 644 631 a(Sc) m(h) m(w) m(artz) p 1054 631 a(distribution) p 1581 631 a(space\),) p 1903 631 a(where) p Fn 2181 631 a(B) p Fo 2260 631 a(,) p Fn 2316 631 a(\013) p Fm 2378 646 a(j) p Fo 2442 631 a(are) p 2600 631 a(constan) m(ts) p 3028 631 a(satisfying) p Fn 3458 631 a(B) p 3565 631 a(>) p Fo 3668 631 a(0) p 3745 631 a(and) 1224 851 y(0) p Fn 1301 851 a(<) p 1404 851 a(\013) p Fm 1466 866 a(j) p Fn 1530 851 a(<) p Fo 1634 851 a(1) p 1715 851 a(for) p 1864 851 a(ev) m(ery) p Fn 2123 851 a(j) p Fo 2197 851 a(=) p 2300 851 a(1) p Fn(;) p 2393 851 a(:) p 2437 851 a(:) p 2481 851 a(:) p 2524 851 a(;) p 2568 851 a(N) p 2651 851 a(:) p Fo 3701 851 a(\(1.3\)) 0 1071 y(The) p 210 1071 a(constan) m(t) p Fn 613 1071 a(B) p Fo 735 1071 a(represen) m(ts) p 1203 1071 a(the) p 1381 1071 a(in) m(tensit) m(y) p 1790 1071 a(of) p 1911 1071 a(a) p 2001 1071 a(uniform) p 2377 1071 a(magnetic) p 2804 1071 a(\014eld.) p 3082 1071 a(The) p 3293 1071 a(constan) m(t) p 3696 1071 a(2) p Fn(\031) t(\013) p Fm 3866 1086 a(j) p Fo 0 1191 a(represen) m(ts) p 463 1191 a(the) p 636 1191 a(magnetic) p 1057 1191 a(\015ux) p 1254 1191 a(of) p 1370 1191 a(an) p 1510 1191 a(in\014nitesimally) p 2146 1191 a(thin) p 2356 1191 a(solenoid) p 2735 1191 a(placed) p 3043 1191 a(at) p Fn 3166 1191 a(z) p Fm 3211 1206 a(j) p Fo 3248 1191 a(.) p 3332 1191 a(W) p 3424 1191 a(e) p 3504 1191 a(will) p 3693 1191 a(sho) m(w) 0 1312 y(that) p 210 1312 a(the) p 376 1312 a(di\013erence) p 811 1312 a(of) p 921 1312 a(in) m(teger) p 1242 1312 a(magnetic) p 1657 1312 a(\015uxes) p 1930 1312 a(can) p 2107 1312 a(b) s(e) p 2239 1312 a(gauged) p 2568 1312 a(out) p 2740 1312 a(b) m(y) p 2874 1312 a(a) p 2953 1312 a(suitable) p 3316 1312 a(unitary) p 3658 1312 a(gauge) 0 1432 y(transform) p 459 1432 a(\(see) p 665 1432 a(Prop) s(osition) p 1201 1432 a(2.3) p 1369 1432 a(b) s(elo) m(w\).) p 1754 1432 a(Since) p 2020 1432 a(w) m(e) p 2174 1432 a(consider) p 2565 1432 a(only) p 2790 1432 a(the) p 2969 1432 a(sp) s(ectral) p 3346 1432 a(problem,) p 3767 1432 a(the) 0 1553 y(assumption) p 513 1553 a(\(1.3\)) p 744 1553 a(loses) p 970 1553 a(no) p 1103 1553 a(generalit) m(y) p 1513 1553 a(.) p 1582 1553 a(W) p 1674 1553 a(e) p 1748 1553 a(\014nd) p 1940 1553 a(a) p 2019 1553 a(pro) s(of) p 2271 1553 a(of) p 2380 1553 a(the) p 2546 1553 a(existence) p 2958 1553 a(of) p 3067 1553 a(the) p 3232 1553 a(v) m(ector) p 3523 1553 a(p) s(oten) m(tial) 0 1673 y(with) p Fn 235 1673 a(\016) p Fo 282 1673 a(-lik) m(e) p 506 1673 a(singularities) p 1062 1673 a(in) p 1188 1673 a(Arai's) p 1486 1673 a(pap) s(er) p 1772 1673 a(\(see) p 1980 1673 a([Ar1]) p 2240 1673 a(and) p 2442 1673 a([Ar2];) p 2735 1673 a(see) p 2905 1673 a(also) p 3113 1673 a(Prop) s(osition) p 3651 1673 a(2.1) p 3821 1673 a(in) 0 1793 y(section) p 332 1793 a(2) p 420 1793 a(b) s(elo) m(w\).) p 792 1793 a(When) p Fn 1083 1793 a(N) p Fo 1211 1793 a(=) p 1326 1793 a(1) p 1413 1793 a(and) p Fn 1610 1793 a(\013) p Fl 1672 1808 a(1) p Fo 1750 1793 a(=) p Fn 1865 1793 a(\013) p Fo 1928 1793 a(,) p 1995 1793 a(w) m(e) p 2146 1793 a(alw) m(a) m(ys) p 2464 1793 a(assume) p 2808 1793 a(that) p Fn 3026 1793 a(z) p Fl 3071 1808 a(1) p Fo 3150 1793 a(=) p 3264 1793 a(0) p 3352 1793 a(and) p 3549 1793 a(tak) m(e) p 3767 1793 a(the) 0 1914 y(circular) p 352 1914 a(gauge,) p 656 1914 a(that) p 867 1914 a(is,) p Fj 1132 2190 a(a) p Fl 1195 2205 a(1) p Fo 1235 2190 a(\() p Fn(z) p Fo 1322 2190 a(\)) p 1388 2190 a(=) p Fi 1491 2044 a( ) p Fk 1557 2190 a(\000) p Fn 1644 2123 a(B) p 1644 2167 80 4 v Fo 1659 2259 a(2) p Fn 1734 2190 a(y) p Fk 1807 2190 a(\000) p Fn 1958 2123 a(\013) p 1917 2167 145 4 v Fk 1917 2259 a(j) p Fn(z) p Fk 1994 2259 a(j) p Fl 2022 2230 a(2) p Fn 2071 2190 a(y) t(;) 2176 2123 y(B) p 2176 2167 80 4 v Fo 2191 2259 a(2) p Fn 2265 2190 a(x) p Fo 2343 2190 a(+) p Fn 2491 2123 a(\013) p 2451 2167 145 4 v Fk 2451 2259 a(j) p Fn(z) p Fk 2528 2259 a(j) p Fl 2556 2230 a(2) p Fn 2605 2190 a(x) p Fi 2660 2044 a(!) p Fn 2743 2190 a(:) p Fo 3701 2190 a(\(1.4\)) 0 2473 y(When) p 284 2473 a(w) m(e) p 428 2473 a(need) p 655 2473 a(to) p 774 2473 a(indicate) p 1142 2473 a(the) p 1310 2473 a(v) p 1356 2473 a(alue) p Fn 1562 2473 a(\013) p Fo 1657 2473 a(explicitly) p 2041 2473 a(,) p 2099 2473 a(w) m(e) p 2242 2473 a(denote) p Fk 2556 2473 a(L) p Fm 2625 2437 a(\013) p Fl 2625 2497 a(1) p Fo 2707 2473 a(for) p Fk 2855 2473 a(L) p Fl 2924 2488 a(1) p Fo 2996 2473 a(\(this) p 3224 2473 a(notation) p 3614 2473 a(is) p 3712 2473 a(used) 0 2593 y(for) p 149 2593 a(the) p 317 2593 a(op) s(erator) p Fn 710 2593 a(L) p Fl 776 2608 a(1) p Fo 848 2593 a(de\014ned) p 1184 2593 a(b) s(elo) m(w\).) 146 2713 y(De\014ne) p 448 2713 a(a) p 530 2713 a(linear) p 801 2713 a(op) s(erator) p Fn 1194 2713 a(L) p Fm 1260 2728 a(N) p Fo 1360 2713 a(on) p Fn 1495 2713 a(L) p Fl 1561 2677 a(2) p Fo 1601 2713 a(\() p Fj(R) p Fl 1726 2671 a(2) p Fo 1766 2713 a(\)) p 1836 2713 a(b) m(y) p Fn 1055 2933 a(L) p Fm 1121 2948 a(N) p Fn 1188 2933 a(u) p Fo 1272 2933 a(=) p Fk 1375 2933 a(L) p Fm 1444 2948 a(N) p Fn 1511 2933 a(u;) p 1643 2933 a(u) p Fk 1726 2933 a(2) p Fn 1820 2933 a(D) p Fo 1904 2933 a(\() p Fn(L) p Fm 2008 2948 a(N) p Fo 2075 2933 a(\)) p 2141 2933 a(=) p Fn 2244 2933 a(C) p Fg 2321 2892 a(1) p Fl 2314 2958 a(0) p Fo 2396 2933 a(\() p Fj(R) p Fl 2521 2891 a(2) p Fk 2583 2933 a(n) p Fn 2655 2933 a(S) p Fm 2715 2948 a(N) p Fo 2782 2933 a(\)) p Fn(;) p Fo 0 3153 a(where) p Fn 284 3153 a(D) p Fo 368 3153 a(\() p Fn(L) p Fo(\)) p 544 3153 a(is) p 644 3153 a(the) p 814 3153 a(op) s(erator) p 1208 3153 a(domain) p 1557 3153 a(of) p 1670 3153 a(a) p 1753 3153 a(linear) p 2026 3153 a(op) s(erator) p Fn 2420 3153 a(L) p Fo 2521 3153 a(and) p Fn 2712 3153 a(C) p Fg 2789 3117 a(1) p Fl 2782 3178 a(0) p Fo 2864 3153 a(\() p Fn(U) p Fo 2978 3153 a(\)) p 3051 3153 a(is) p 3151 3153 a(the) p 3321 3153 a(space) p 3583 3153 a(of) p 3696 3153 a(com-) 0 3274 y(pactly) p 289 3274 a(supp) s(orted) p 740 3274 a(smo) s(oth) p 1078 3274 a(functions) p 1492 3274 a(in) p 1599 3274 a(an) p 1728 3274 a(op) s(en) p 1956 3274 a(set) p Fn 2102 3274 a(U) p Fo 2178 3274 a(.) p 2247 3274 a(The) p 2440 3274 a(op) s(erator) p Fn 2827 3274 a(L) p Fm 2893 3289 a(N) p Fo 2986 3274 a(is) p 3077 3274 a(symmetric,) p 3574 3274 a(p) s(ositiv) m(e) 0 3394 y(and) p 190 3394 a(has) p 365 3394 a(the) p 534 3394 a(de\014ciency) p 981 3394 a(indices) p 1302 3394 a(\(2) p Fn(N) p 1472 3394 a(;) p Fo 1516 3394 a(2) p Fn(N) p Fo 1653 3394 a(\)) p 1724 3394 a(\(see) p 1921 3394 a(Prop) s(osition) p 2446 3394 a(4.5) p 2604 3394 a(b) s(elo) m(w\).) p 2958 3394 a(Th) m(us) p 3206 3394 a(the) p 3375 3394 a(op) s(erator) p Fn 3768 3394 a(L) p Fm 3834 3409 a(N) p Fo 0 3515 a(has) p 173 3515 a(self-adjoin) m(t) p 674 3515 a(extensions) p 1143 3515 a(parameterized) p 1776 3515 a(b) m(y) p 1910 3515 a(\(2) p Fn(N) p Fk 2105 3515 a(\002) p Fo 2203 3515 a(2) p Fn(N) p Fo 2340 3515 a(\)-unitary) p 2754 3515 a(matrices) p 3143 3515 a(\(see) p 3338 3515 a([Re-Si,) p 3653 3515 a(Theo-) 0 3635 y(rem) p 196 3635 a(X.2]\).) p 484 3635 a(W) p 576 3635 a(e) p 654 3635 a(denote) p 969 3635 a(one) p 1149 3635 a(of) p 1261 3635 a(self-adjoin) m(t) p 1764 3635 a(extensions) p 2235 3635 a(of) p Fn 2348 3635 a(L) p Fm 2414 3650 a(N) p Fo 2515 3635 a(b) m(y) p Fn 2652 3635 a(H) p Fm 2733 3650 a(N) p Fo 2800 3635 a(.) p 2874 3635 a(In) p 2997 3635 a(particular,) p 3475 3635 a(w) m(e) p 3620 3635 a(denote) 0 3755 y(the) p 175 3755 a(F) p 231 3755 a(riedric) m(hs) p 631 3755 a(extension) p 1069 3755 a(of) p Fn 1187 3755 a(L) p Fm 1253 3770 a(N) p Fo 1360 3755 a(\(the) p 1573 3755 a(self-adjoin) m(t) p 2082 3755 a(op) s(erator) p 2481 3755 a(asso) s(ciated) p 2953 3755 a(to) p 3079 3755 a(the) p 3254 3755 a(form) p 3491 3755 a(closure) p 3823 3755 a(of) p Fn 0 3876 a(D) p Fo 84 3876 a(\() p Fn(L) p Fm 188 3891 a(N) p Fo 255 3876 a(\),) p 357 3876 a(see) p 518 3876 a([Re-Si,) p 838 3876 a(Theorem) p 1253 3876 a(X.23]\)) p 1552 3876 a(b) m(y) p Fn 1691 3876 a(H) p Fm 1780 3840 a(AB) 1772 3900 y(N) p Fo 1893 3876 a(,) p 1956 3876 a(whic) m(h) p 2239 3876 a(is) p 2340 3876 a(called) p 2620 3876 a(the) p 2791 3876 a(standard) p 3202 3876 a(Aharono) m(v-Bohm) 0 3996 y(Hamiltonian) p 561 3996 a(\(this) p 789 3996 a(name) p 1049 3996 a(is) p 1147 3996 a(used) p 1370 3996 a(in) p 1484 3996 a([Ex-St-Vy],) p 1997 3996 a(when) p Fn 2252 3996 a(N) p Fo 2368 3996 a(=) p 2472 3996 a(1\).) 146 4117 y(The) p 347 4117 a(Sc) m(hr\177) p 533 4117 a(odinger) p 881 4117 a(op) s(erator) p 1274 4117 a(with) p 1496 4117 a(constan) m(t) p 1889 4117 a(magnetic) p 2307 4117 a(\014eld) p 2518 4117 a(is) p 2616 4117 a(giv) m(en) p 2871 4117 a(b) m(y) p Fk 1104 4392 a(L) p Fl 1173 4407 a(0) p Fo 1240 4392 a(=) p Fi 1344 4271 a(\022) p Fo 1415 4325 a(1) p 1415 4369 49 4 v Fn 1423 4460 a(i) p Fk 1473 4392 a(r) p Fo 1578 4392 a(+) p Fj 1676 4392 a(a) p Fl 1740 4407 a(0) p Fi 1779 4271 a(\023) p Fl 1840 4287 a(2) p Fn 1896 4392 a(;) p Fj 1972 4392 a(a) p Fl 2035 4407 a(0) p Fo 2103 4392 a(=) p Fi 2206 4271 a(\022) p Fk 2267 4392 a(\000) p Fn 2354 4325 a(B) p 2354 4369 80 4 v Fo 2369 4460 a(2) p Fn 2444 4392 a(y) t(;) 2549 4325 y(B) p 2549 4369 V Fo 2564 4460 a(2) p Fn 2638 4392 a(x) p Fi 2693 4271 a(\023) p Fn 2771 4392 a(:) p Fo 3701 4392 a(\(1.5\)) 0 4644 y(It) p 106 4644 a(is) p 204 4644 a(w) m(ell-kno) m(wn) p 710 4644 a(that) p 922 4644 a(the) p 1090 4644 a(linear) p 1361 4644 a(op) s(erator) p 1754 4644 a(de\014ned) p 2090 4644 a(b) m(y) p Fn 1309 4864 a(L) p Fl 1375 4879 a(0) p Fn 1415 4864 a(u) p Fo 1498 4864 a(=) p Fk 1602 4864 a(L) p Fl 1671 4879 a(0) p Fn 1710 4864 a(u;) p 1842 4864 a(D) p Fo 1926 4864 a(\() p Fn(L) p Fl 2030 4879 a(0) p Fo 2069 4864 a(\)) p 2135 4864 a(=) p Fn 2238 4864 a(C) p Fg 2315 4823 a(1) p Fl 2308 4888 a(0) p Fo 2390 4864 a(\() p Fj(R) p Fl 2515 4822 a(2) p Fo 2555 4864 a(\)) p 3701 4864 a(\(1.6\)) 1927 5214 y(2) p eop %%Page: 3 3 3 2 bop Fo 0 631 a(is) p 110 631 a(essen) m(tially) p 589 631 a(self-adjoin) m(t) p 1102 631 a(and) p 1304 631 a(the) p 1483 631 a(sp) s(ectrum) p 1921 631 a(of) p 2044 631 a(the) p 2224 631 a(unique) p 2552 631 a(self-adjoin) m(t) p 3066 631 a(extension) p Fn 3509 631 a(H) p Fl 3590 646 a(0) p Fo 3673 631 a(of) p Fn 3796 631 a(L) p Fl 3862 646 a(0) p Fo 0 751 a(satis\014es) p Fn 1212 872 a(\033) p Fo 1271 872 a(\() p Fn(H) p Fl 1390 887 a(0) p Fo 1429 872 a(\)) p 1495 872 a(=) p Fk 1598 872 a(f) p Fo(\(2) p Fn(n) p Fk 1815 872 a(\000) p Fo 1915 872 a(1\)) p Fn(B) p Fo 2081 872 a(;) p Fn 2125 872 a(n) p Fo 2210 872 a(=) p 2314 872 a(1) p Fn(;) p Fo 2407 872 a(2) p Fn(;) p 2500 872 a(:) p 2544 872 a(:) p 2588 872 a(:) p Fk 2613 872 a(g) p Fn(;) p Fo 0 1046 a(whic) m(h) p 279 1046 a(is) p 377 1046 a(called) p 654 1046 a(the) p 822 1046 a(Landau) p 1175 1046 a(lev) m(els.) 146 1166 y(When) p 435 1166 a(solenoids) p 853 1166 a(exist,) p 1117 1166 a(the) p 1289 1166 a(sp) s(ectrum) p 1720 1166 a(in) p 1838 1166 a(a) p 1924 1166 a(gap) p 2113 1166 a(of) p 2228 1166 a(the) p 2401 1166 a(Landau) p 2759 1166 a(lev) m(els) p 3024 1166 a(app) s(ears.) p 3437 1166 a(Our) p 3642 1166 a(aim) p 3836 1166 a(is) 0 1287 y(to) p 123 1287 a(giv) m(e) p 328 1287 a(an) p 468 1287 a(estimate) p 863 1287 a(for) p 1016 1287 a(the) p 1188 1287 a(n) m(um) m(b) s(er) p 1547 1287 a(of) p 1662 1287 a(eigen) m(v) p 1922 1287 a(alues) p 2171 1287 a(b) s(et) m(w) m(een) p 2552 1287 a(t) m(w) m(o) p 2740 1287 a(Landau) p 3098 1287 a(lev) m(els) p 3363 1287 a(or) p 3486 1287 a(b) s(elo) m(w) p 3767 1287 a(the) 0 1407 y(lo) m(w) m(est) p 293 1407 a(Landau) p 647 1407 a(lev) m(el.) 146 1527 y(W) p 238 1527 a(e) p 304 1527 a(recall) p 554 1527 a(kno) m(wn) p 853 1527 a(results) p 1153 1527 a(in) p 1257 1527 a(the) p 1414 1527 a(case) p Fn 1611 1527 a(N) p Fo 1727 1527 a(=) p 1831 1527 a(1.) p 1946 1527 a(Nam) m(bu) p 2278 1527 a(\([Nam]\)) p 2633 1527 a(treats) p 2900 1527 a(the) p 3058 1527 a(standard) p 3455 1527 a(Aharono) m(v-) 0 1648 y(Bohm) p 293 1648 a(Hamiltonian) p Fn 861 1648 a(H) p Fm 950 1612 a(AB) p Fl 942 1672 a(1) p Fo 1103 1648 a(and) p 1299 1648 a(giv) m(es) p 1545 1648 a(an) p 1688 1648 a(explicit) p 2039 1648 a(represen) m(tation) p 2684 1648 a(of) p 2802 1648 a(all) p 2944 1648 a(eigen) m(v) p 3204 1648 a(alues) p 3456 1648 a(and) p 3653 1648 a(eigen-) 0 1768 y(functions) p 425 1768 a(using) p 685 1768 a(complex) p 1072 1768 a(in) m(tegration) p 1572 1768 a(\(he) p 1745 1768 a(treats) p 2026 1768 a(also) p 2226 1768 a(the) p 2399 1768 a(case) p Fn 2610 1768 a(B) p Fo 2725 1768 a(=) p 2836 1768 a(0\).) p 3007 1768 a(Exner,) 3328 1743 y(\024) 3326 1768 y(St'o) m(v) p 3531 1768 a(\023) p 3542 1768 a(\020) m(\024) p 3570 1768 a(cek) p 3745 1768 a(and) 0 1888 y(Vyt) p 157 1888 a(\024) p 163 1888 a(ras) p 317 1888 a(\([Ex-St-Vy]\)) p 876 1888 a(giv) m(e) p 1074 1888 a(a) p 1152 1888 a(detailed) p 1518 1888 a(analysis) p 1881 1888 a(for) p 2027 1888 a(ev) m(ery) p 2282 1888 a(self-adjoin) m(t) p 2781 1888 a(extension) p Fn 3209 1888 a(H) p Fl 3290 1903 a(1) p Fo 3329 1888 a(.) p 3399 1888 a(In) p 3518 1888 a(the) p 3683 1888 a(p) s(olar) 0 2009 y(co) s(ordinate) p Fn 480 2009 a(x) p Fo 557 2009 a(+) p Fn 655 2009 a(iy) p Fo 767 2009 a(=) p Fn 871 2009 a(r) s(e) p Fm 963 1973 a(i\022) p Fo 1026 2009 a(,) p 1085 2009 a(the) p 1253 2009 a(op) s(erator) p Fn 1646 2009 a(H) p Fl 1727 2024 a(1) p Fo 1799 2009 a(tak) m(es) p 2049 2009 a(the) p 2217 2009 a(form) p Fn 1076 2287 a(H) p Fl 1157 2302 a(1) p Fo 1223 2287 a(=) p Fk 1327 2287 a(\000) p Fo 1414 2220 a(1) p 1414 2264 49 4 v Fn 1415 2356 a(r) 1473 2287 y(@) p Fm 1524 2302 a(r) p Fo 1563 2287 a(\() p Fn(r) s(@) p Fm 1699 2302 a(r) p Fo 1737 2287 a(\)) p 1797 2287 a(+) 1923 2220 y(1) p 1905 2264 87 4 v Fn 1905 2356 a(r) p Fl 1952 2327 a(2) p Fi 2017 2166 a(\022) p Fo 2089 2220 a(1) p 2089 2264 49 4 v Fn 2097 2356 a(i) 2147 2287 y(@) p Fm 2198 2302 a(\022) p Fo 2260 2287 a(+) p Fn 2358 2287 a(\013) p Fo 2443 2287 a(+) p Fn 2551 2220 a(B) p 2551 2264 80 4 v Fo 2566 2356 a(2) p Fn 2640 2287 a(r) p Fl 2687 2246 a(2) p Fi 2726 2166 a(\023) p Fl 2787 2178 a(2) p Fo 3701 2287 a(\(1.7\)) 0 2557 y(if) p 95 2557 a(one) p 280 2557 a(tak) m(es) p 536 2557 a(the) p 709 2557 a(circular) p 1068 2557 a(gauge) p 1350 2557 a(\(1.4\).) p 1638 2557 a(Exner,) 1961 2532 y(\024) 1959 2557 y(St'o) m(v) p 2164 2557 a(\023) p 2175 2557 a(\020) m(\024) p 2203 2557 a(cek) p 2379 2557 a(and) p 2575 2557 a(Vyt) p 2732 2557 a(\024) p 2738 2557 a(ras) p 2901 2557 a(solv) m(e) p 3146 2557 a(the) p 3319 2557 a(eigenequation) 0 2678 y(explicitly) p 421 2678 a(b) m(y) p 555 2678 a(using) p 809 2678 a(F) p 865 2678 a(ourier) p 1145 2678 a(expansion) p 1596 2678 a(with) p 1817 2678 a(resp) s(ect) p 2150 2678 a(to) p 2267 2678 a(the) p Fn 2434 2678 a(\022) p Fo 2482 2678 a(-v) p 2561 2678 a(ariable) p 2879 2678 a(and) p 3067 2678 a(represen) m(t) p 3486 2678 a(the) p 3653 2678 a(eigen-) 0 2798 y(v) p 46 2798 a(alue) p 247 2798 a(as) p 361 2798 a(a) p 437 2798 a(solution) p 801 2798 a(of) p 907 2798 a(an) p 1037 2798 a(equation) p 1430 2798 a(including) p 1847 2798 a(Gamma) p 2211 2798 a(functions.) p 2668 2798 a(W) p 2760 2798 a(e) p 2831 2798 a(summarize) p 3314 2798 a(a) p 3390 2798 a(part) p 3596 2798 a(of) p 3701 2798 a(their) 0 2918 y(results) p 310 2918 a(as) p 430 2918 a(follo) m(ws.) p Fq 146 3121 a(Theorem) p 621 3121 a(1.1) p 802 3121 a(\(Nam) m(bu,) p 1271 3121 a(Exner-) 1603 3097 y(\024) 1600 3121 y(St'o) m(v) p 1837 3121 a(\023) p 1849 3121 a(\020) m(\024) p 1881 3121 a(cek-Vyt) p 2261 3121 a(\024) p 2266 3121 a(ras\)) p Ff 2502 3121 a(\(i\)) p 2648 3121 a(The) p 2849 3121 a(sp) p 2934 3121 a(e) p 2974 3121 a(ctrum) p 3260 3121 a(of) p 3376 3121 a(the) p 3540 3121 a(standar) p 3853 3121 a(d) 0 3241 y(A) n(har) p 206 3241 a(onov-Bohm) p 723 3241 a(Hamiltonian) p Fn 1286 3241 a(H) p Fm 1375 3205 a(AB) p Fl 1367 3266 a(1) p Ff 1523 3241 a(is) p 1627 3241 a(given) p 1881 3241 a(by) p Fn 437 3461 a(\033) p Fo 496 3461 a(\() p Fn(H) p Fm 623 3420 a(AB) p Fl 615 3485 a(1) p Fo 736 3461 a(\)) p 802 3461 a(=) p Fk 905 3461 a(f) p Fo(\(2) p Fn(n) p Fk 1122 3461 a(\000) p Fo 1222 3461 a(1\)) p Fn(B) p Fo 1388 3461 a(;) p Fn 1432 3461 a(n) p Fo 1517 3461 a(=) p 1621 3461 a(1) p Fn(;) p Fo 1714 3461 a(2) p Fn(;) p 1807 3461 a(:) p 1851 3461 a(:) p 1895 3461 a(:) p 1937 3461 a(;) p Fk 1981 3461 a(g) p 2053 3461 a([) p 2141 3461 a(f) p Fo(\(2) p Fn(n) p Fo 2358 3461 a(+) p 2456 3461 a(2) p Fn(\013) p Fk 2590 3461 a(\000) p Fo 2689 3461 a(1\)) p Fn(B) p Fo 2855 3461 a(;) p Fn 2899 3461 a(n) p Fo 2985 3461 a(=) p 3088 3461 a(1) p Fn(;) p Fo 3181 3461 a(2) p Fn(;) p 3274 3461 a(:) p 3318 3461 a(:) p 3362 3461 a(:) p Fk 3388 3461 a(g) p Fn(:) p Ff 0 3680 a(The) p 199 3680 a(multiplicity) p 712 3680 a(of) p 827 3680 a(e) p 867 3680 a(ach) p 1046 3680 a(eigenvalue) p 1517 3680 a(is) p 1621 3680 a(given) p 1875 3680 a(by) p Fo 1106 3899 a(m) m(ult) o(\(\(2) p Fn(n) p Fk 1509 3899 a(\000) p Fo 1609 3899 a(1\)) p Fn(B) p Fo 1775 3899 a(;) p Fn 1819 3899 a(H) p Fm 1908 3858 a(AB) p Fl 1900 3924 a(1) p Fo 2020 3899 a(\)) p 2141 3899 a(=) p Fk 2300 3899 a(1) p Fn(;) p 2478 3899 a(n) p Fo 2564 3899 a(=) p 2668 3899 a(1) p Fn(;) p Fo 2761 3899 a(2) p Fn(;) p 2854 3899 a(:) p 2898 3899 a(:) p 2942 3899 a(:) p 2984 3899 a(;) p Fo 875 4045 a(m) m(ult) n(\(\(2) p Fn(n) p Fo 1277 4045 a(+) p 1375 4045 a(2) p Fn(\013) p Fk 1509 4045 a(\000) p Fo 1609 4045 a(1\)) p Fn(B) p Fo 1775 4045 a(;) p Fn 1819 4045 a(H) p Fm 1908 4003 a(AB) p Fl 1900 4069 a(1) p Fo 2020 4045 a(\)) p 2141 4045 a(=) p Fn 2300 4045 a(n;) p 2437 4045 a(n) p Fo 2523 4045 a(=) p 2626 4045 a(1) p Fn(;) p Fo 2719 4045 a(2) p Fn(;) p 2812 4045 a(:) p 2856 4045 a(:) p 2900 4045 a(:) p 2942 4045 a(;) p Ff 0 4264 a(wher) p 196 4264 a(e) p Fo 275 4264 a(m) m(ult) o(\() p Fn(\025) p Fo(;) p Fn 612 4264 a(H) p Fo 701 4264 a(\)) p Ff 773 4264 a(is) p 878 4264 a(the) p 1039 4264 a(multiplicity) p 1552 4264 a(of) p 1667 4264 a(an) p 1807 4264 a(eigenvalue) p Fn 2277 4264 a(\025) p Ff 2369 4264 a(of) p 2484 4264 a(a) p 2568 4264 a(self-adjoint) p 3074 4264 a(op) p 3169 4264 a(er) p 3250 4264 a(ator) p Fn 3457 4264 a(H) p Ff 3546 4264 a(.) 146 4384 y(\(ii\)) p Fn 321 4384 a(L) p Fl 387 4348 a(2) p Fo 427 4384 a(\() p Fj(R) p Fl 552 4342 a(2) p Fo 591 4384 a(\)) p Ff 664 4384 a(is) p 769 4384 a(de) p 859 4384 a(c) p 899 4384 a(omp) p 1074 4384 a(ose) p 1204 4384 a(d) p 1287 4384 a(into) p 1489 4384 a(the) p 1652 4384 a(dir) p 1768 4384 a(e) p 1808 4384 a(ct) p 1920 4384 a(sum) p 2126 4384 a(of) p 2241 4384 a(two) p 2423 4384 a(close) p 2623 4384 a(d) p 2707 4384 a(subsp) p 2929 4384 a(ac) p 3019 4384 a(es) p Fk 3138 4384 a(H) p Fm 3222 4399 a(s) p Ff 3294 4384 a(and) p Fk 3484 4384 a(H) p Fm 3568 4399 a(c) p Ff 3603 4384 a(,) p 3668 4384 a(c) p 3708 4384 a(al) p 3788 4384 a(le) p 3853 4384 a(d) 0 4505 y(the) p 165 4505 a(stable) p 439 4505 a(subsp) p 661 4505 a(ac) p 751 4505 a(e) p 833 4505 a(and) p 1026 4505 a(critic) p 1244 4505 a(al) p 1356 4505 a(subsp) p 1578 4505 a(ac) p 1668 4505 a(e,) p 1781 4505 a(r) p 1817 4505 a(esp) p 1947 4505 a(e) p 1987 4505 a(ctively.) p 2339 4505 a(The) p 2541 4505 a(sp) p 2626 4505 a(ac) p 2716 4505 a(es) p Fk 2838 4505 a(H) p Fm 2922 4520 a(s) p Ff 2997 4505 a(and) p Fk 3189 4505 a(H) p Fm 3273 4520 a(c) p Ff 3346 4505 a(ar) p 3432 4505 a(e) p 3515 4505 a(invariant) 0 4625 y(subsp) p 222 4625 a(ac) p 312 4625 a(es) p 430 4625 a(for) p 585 4625 a(any) p 771 4625 a(self-adjoint) p 1276 4625 a(extension) p Fn 1706 4625 a(H) p Fl 1787 4640 a(1) p Ff 1860 4625 a(of) p Fn 1974 4625 a(L) p Fl 2040 4640 a(1) p Ff 2079 4625 a(.) p 2154 4625 a(The) p 2352 4625 a(r) p 2388 4625 a(estricte) p 2693 4625 a(d) p 2777 4625 a(op) p 2872 4625 a(er) p 2953 4625 a(ator) p Fn 3160 4625 a(H) p Fl 3241 4640 a(1) p Fk 3280 4625 a(j) p Fg 3308 4640 a(H) p Fe 3368 4648 a(s) p Ff 3440 4625 a(is) p 3543 4625 a(indep) p 3768 4625 a(en-) 0 4745 y(dent) p 217 4745 a(of) p 331 4745 a(the) p 493 4745 a(choic) p 708 4745 a(e) p 787 4745 a(of) p Fn 902 4745 a(H) p Fl 983 4760 a(1) p Ff 1057 4745 a(and) p 1246 4745 a(the) p 1408 4745 a(sp) p 1493 4745 a(e) p 1533 4745 a(ctrum) p 1818 4745 a(of) p Fn 1932 4745 a(H) p Fl 2013 4760 a(1) p Fk 2052 4745 a(j) p Fg 2080 4760 a(H) p Fe 2140 4768 a(s) p Ff 2213 4745 a(is) p 2317 4745 a(given) p 2571 4745 a(by) p Fn 437 4965 a(\033) p Fo 513 4965 a(\() p Fn(H) p Fl 632 4980 a(1) p Fk 671 4965 a(j) p Fg 699 4980 a(H) p Fe 759 4988 a(s) p Fo 796 4965 a(\)) p 862 4965 a(=) p Fk 966 4965 a(f) p Fo(\(2) p Fn(n) p Fk 1182 4965 a(\000) p Fo 1282 4965 a(1\)) p Fn(B) p Fo 1448 4965 a(;) p Fn 1492 4965 a(n) p Fo 1578 4965 a(=) p 1681 4965 a(1) p Fn(;) p Fo 1774 4965 a(2) p Fn(;) p 1867 4965 a(:) p 1911 4965 a(:) p 1955 4965 a(:) p Fk 1981 4965 a(g) p 2052 4965 a([) p 2141 4965 a(f) p Fo(\(2) p Fn(n) p Fo 2358 4965 a(+) p 2456 4965 a(2) p Fn(\013) p Fk 2590 4965 a(\000) p Fo 2689 4965 a(1\)) p Fn(B) p Fo 2855 4965 a(;) p Fn 2899 4965 a(n) p Fo 2985 4965 a(=) p 3088 4965 a(2) p Fn(;) p Fo 3181 4965 a(3) p Fn(;) p 3274 4965 a(:) p 3318 4965 a(:) p 3362 4965 a(:) p Fk 3388 4965 a(g) p Fn(:) p Fo 1927 5214 a(3) p eop %%Page: 4 4 4 3 bop Ff 0 631 a(The) p 199 631 a(multiplicity) p 712 631 a(of) p 827 631 a(e) p 867 631 a(ach) p 1046 631 a(eigenvalue) p 1517 631 a(is) p 1621 631 a(given) p 1875 631 a(by) p Fo 1028 846 a(m) m(ult) o(\(\(2) p Fn(n) p Fk 1431 846 a(\000) p Fo 1531 846 a(1\)) p Fn(B) p Fo 1697 846 a(;) p Fn 1741 846 a(H) p Fl 1822 861 a(1) p Fk 1860 846 a(j) p Fg 1888 861 a(H) p Fe 1948 869 a(s) p Fo 1986 846 a(\)) p 2107 846 a(=) p Fk 2265 846 a(1) p Fn(;) p 2444 846 a(n) p Fo 2530 846 a(=) p 2633 846 a(1) p Fn(;) p Fo 2726 846 a(2) p Fn(;) p 2819 846 a(:) p 2863 846 a(:) p 2907 846 a(:) p 2949 846 a(;) p Fo 797 991 a(m) m(ult) n(\(\(2) p Fn(n) p Fo 1199 991 a(+) p 1297 991 a(2) p Fn(\013) p Fk 1431 991 a(\000) p Fo 1531 991 a(1\)) p Fn(B) p Fo 1697 991 a(;) p Fn 1741 991 a(H) p Fl 1822 1006 a(1) p Fk 1860 991 a(j) p Fg 1888 1006 a(H) p Fe 1948 1014 a(s) p Fo 1986 991 a(\)) p 2107 991 a(=) p Fn 2265 991 a(n) p Fk 2346 991 a(\000) p Fo 2445 991 a(1) p Fn(;) p 2573 991 a(n) p Fo 2659 991 a(=) p 2762 991 a(2) p Fn(;) p Fo 2855 991 a(3) p Fn(;) p 2948 991 a(:) p 2992 991 a(:) p 3036 991 a(:) p 3078 991 a(:) p Ff 0 1207 a(The) p 200 1207 a(r) p 236 1207 a(estricte) p 541 1207 a(d) p 628 1207 a(op) p 723 1207 a(er) p 804 1207 a(ator) p Fn 1012 1207 a(H) p Fl 1093 1222 a(1) p Fk 1133 1207 a(j) p Fg 1161 1222 a(H) p Fe 1221 1230 a(c) p Ff 1293 1207 a(dep) p 1433 1207 a(ends) p 1657 1207 a(on) p 1798 1207 a(the) p 1961 1207 a(choic) p 2176 1207 a(e) p 2256 1207 a(of) p 2372 1207 a(self-adjoint) p 2879 1207 a(extension) p Fn 3311 1207 a(H) p Fl 3392 1222 a(1) p Ff 3431 1207 a(.) p 3510 1207 a(However,) 0 1327 y(the) p 162 1327 a(fol) p 272 1327 a(lowing) p 575 1327 a(estimates) p 1004 1327 a(hold) p 1213 1327 a(indep) p 1438 1327 a(endently) p 1826 1327 a(of) p 1940 1327 a(the) p 2102 1327 a(choic) p 2317 1327 a(e) p 2396 1327 a(of) p Fn 2511 1327 a(H) p Fl 2592 1342 a(1) p Ff 2631 1327 a(.) p Fo 990 1542 a(dim) p 1169 1542 a(Ran) p Fn 1361 1542 a(P) p Fl 1424 1558 a(\() p Fg(\0001) p Fm(;) p Fl(\(2) p Fm(\013) p Fg(\000) p Fl(1\)) p Fm(B) p Fl 1877 1558 a(\)) p Fo 1910 1542 a(\() p Fn(H) p Fl 2029 1557 a(1) p Fk 2068 1542 a(j) p Fg 2096 1557 a(H) p Fe 2156 1565 a(c) p Fo 2192 1542 a(\)) p Fk 2313 1542 a(\024) p Fo 2474 1542 a(2) p Fn(;) p Fo 703 1687 a(dim) p 882 1687 a(Ran) p Fn 1074 1687 a(P) p Fl 1137 1703 a(\(\(2) p Fm(n) p Fl(+2) p Fm(\013) p Fg(\000) p Fl(1\)) p Fm(B) s(;) p Fl(\(2) p Fm(n) p Fl(+1\)) p Fm(B) p Fl 1875 1703 a(\)) p Fo 1910 1687 a(\() p Fn(H) p Fl 2029 1702 a(1) p Fk 2068 1687 a(j) p Fg 2096 1702 a(H) p Fe 2156 1710 a(c) p Fo 2192 1687 a(\)) p Fk 2313 1687 a(\024) p Fo 2474 1687 a(2) p Fn(;) p 2601 1687 a(n) p Fo 2687 1687 a(=) p 2790 1687 a(0) p Fn(;) p Fo 2883 1687 a(1) p Fn(;) p Fo 2976 1687 a(2) p Fn(;) p 3069 1687 a(:) p 3113 1687 a(:) p 3157 1687 a(:) p Fo 703 1833 a(dim) p 882 1833 a(Ran) p Fn 1074 1833 a(P) p Fl 1137 1848 a(\(\(2) p Fm(n) p Fg(\000) p Fl(1\)) p Fm(B) s(;) p Fl(\(2) p Fm(n) p Fl(+2) p Fm(\013) p Fg(\000) p Fl(1\)) p Fm(B) p Fl 1875 1848 a(\)) p Fo 1910 1833 a(\() p Fn(H) p Fl 2029 1848 a(1) p Fk 2068 1833 a(j) p Fg 2096 1848 a(H) p Fe 2156 1856 a(c) p Fo 2192 1833 a(\)) p Fk 2313 1833 a(\024) p Fo 2474 1833 a(2) p Fn(;) p 2601 1833 a(n) p Fo 2687 1833 a(=) p 2790 1833 a(1) p Fn(;) p Fo 2883 1833 a(2) p Fn(;) p 2976 1833 a(:) p 3020 1833 a(:) p 3064 1833 a(:) p Ff 0 2048 a(wher) p 196 2048 a(e) p Fn 272 2048 a(P) p Fm 335 2063 a(I) p Fo 375 2048 a(\() p Fn(H) p Fo 502 2048 a(\)) p Ff 572 2048 a(denotes) p 920 2048 a(the) p 1079 2048 a(sp) p 1164 2048 a(e) p 1204 2048 a(ctr) p 1317 2048 a(al) p 1424 2048 a(pr) p 1510 2048 a(oje) p 1630 2048 a(ction) p 1873 2048 a(of) p 1985 2048 a(a) p 2067 2048 a(self-adjoint) p 2569 2048 a(op) p 2664 2048 a(er) p 2745 2048 a(ator) p Fn 2950 2048 a(H) p Ff 3071 2048 a(c) p 3111 2048 a(orr) p 3238 2048 a(esp) p 3368 2048 a(onding) p 3683 2048 a(to) p 3797 2048 a(an) 0 2168 y(interval) p Fn 356 2168 a(I) p Ff 407 2168 a(.) p 482 2168 a(The) p 680 2168 a(left-hand) p 1085 2168 a(side) p 1283 2168 a(of) p 1396 2168 a(e) p 1436 2168 a(ach) p 1614 2168 a(of) p 1728 2168 a(thr) p 1846 2168 a(e) p 1886 2168 a(e) p 1965 2168 a(ine) p 2090 2168 a(qualities) p 2472 2168 a(ab) p 2562 2168 a(ove) p 2735 2168 a(takes) p 2980 2168 a(the) p 3141 2168 a(values) p Fo 3431 2168 a(0) p Fn(;) p Fo 3524 2168 a(1) p Fn(;) p Fo 3617 2168 a(2) p Ff 3699 2168 a(if) p 3792 2168 a(we) 0 2289 y(take) p 207 2289 a(an) p 346 2289 a(appr) p 532 2289 a(opriate) p 864 2289 a(self-adjoint) p 1370 2289 a(extension) p Fn 1801 2289 a(H) p Fl 1882 2304 a(1) p Ff 1921 2289 a(.) p Fo 0 2488 a(F) p 56 2488 a(rom) p 256 2488 a(Theorem) p 668 2488 a(1.1,) p 852 2488 a(it) p 950 2488 a(follo) m(ws) p 1270 2488 a(that) p Fn 956 2703 a(n) p Fk 1037 2703 a(\000) p Fo 1136 2703 a(1) p Fk 1213 2703 a(\024) p Fo 1318 2703 a(dim) p 1497 2703 a(Ran) p Fn 1689 2703 a(P) p Fl 1752 2719 a(\(\(2) p Fm(n) p Fg(\000) p Fl(1\)) p Fm(B) s(;) p Fl(\(2) p Fm(n) p Fl(+1\)) p Fm(B) p Fl 2355 2719 a(\)) p Fo 2389 2703 a(\() p Fn(H) p Fl 2508 2718 a(1) p Fo 2548 2703 a(\)) p Fk 2613 2703 a(\024) p Fn 2718 2703 a(n) p Fo 2799 2703 a(+) p 2897 2703 a(3) p 3701 2703 a(\(1.8\)) 0 2918 y(for) p Fn 156 2918 a(n) p Fo 253 2918 a(=) p 368 2918 a(1) p Fn(;) p Fo 461 2918 a(2) p Fn(;) p 554 2918 a(:) p 598 2918 a(:) p 642 2918 a(:) p Fo 667 2918 a(,) p 735 2918 a(if) p 831 2918 a(\(2) p Fn(n) p Fo 1003 2918 a(+) p 1106 2918 a(2) p Fn(\013) p Fk 1244 2918 a(\000) p Fo 1348 2918 a(1\)) p Fn(B) p Fo 1553 2918 a(is) p 1658 2918 a(not) p 1838 2918 a(an) p 1980 2918 a(eigen) m(v) p 2240 2918 a(alue) p 2453 2918 a(of) p Fn 2571 2918 a(H) p Fl 2652 2933 a(1) p Fk 2691 2918 a(j) p Fg 2719 2933 a(H) p Fe 2779 2941 a(c) p Fo 2854 2918 a(\(this) p 3089 2918 a(condition) p 3524 2918 a(holds) p 3785 2918 a(for) 0 3039 y(generic) p 335 3039 a(self-adjoin) m(t) p 840 3039 a(extension) p Fn 1276 3039 a(H) p Fl 1357 3054 a(1) p Fo 1396 3039 a(\).) p 1517 3039 a(Later) p 1782 3039 a(w) m(e) p 1930 3039 a(sho) m(w) p 2176 3039 a(that) p 2391 3039 a(the) p 2563 3039 a(upp) s(er) p 2846 3039 a(b) s(ound) p 3151 3039 a(can) p 3334 3039 a(b) s(e) p 3471 3039 a(sharp) s(ened) 0 3159 y(\(see) p 196 3159 a(\(1.12\)) p 477 3159 a(b) s(elo) m(w\).) 146 3279 y(According) p 615 3279 a(to) p 739 3279 a(\(i\)) p 879 3279 a(of) p 995 3279 a(Theorem) p 1412 3279 a(1.1,) p 1602 3279 a(there) p 1856 3279 a(are) p Fn 2024 3279 a(n) p Fo 2119 3279 a(eigenstates) p 2618 3279 a(of) p 2734 3279 a(the) p 2907 3279 a(Hamiltonian) p Fn 3473 3279 a(H) p Fm 3562 3243 a(AB) p Fl 3554 3304 a(1) p Fo 3712 3279 a(with) 0 3400 y(the) p 168 3400 a(energy) p 479 3400 a(b) s(et) m(w) m(een) p Fn 856 3400 a(n) p Fo 946 3400 a(th) p 1071 3400 a(Landau) p 1424 3400 a(lev) m(el) p 1646 3400 a(and) p 1835 3400 a(the) p 2003 3400 a(\() p Fn(n) p Fo 2121 3400 a(+) p 2219 3400 a(1\)) p 2338 3400 a(st) p 2446 3400 a(Landau) p 2800 3400 a(lev) m(el.) p 3060 3400 a(W) p 3152 3400 a(e) p 3227 3400 a(shall) p 3455 3400 a(try) p 3615 3400 a(to) p 3734 3400 a(giv) m(e) 0 3520 y(a) p 81 3520 a(ph) m(ysical) p 456 3520 a(in) m(terpretation) p 1076 3520 a(of) p 1187 3520 a(this) p 1378 3520 a(phenomenon.) 146 3641 y(In) p 270 3641 a(classical) p 646 3641 a(mec) m(hanics,) p 1139 3641 a(an) p 1276 3641 a(electron) p 1646 3641 a(in) p 1762 3641 a(a) p 1845 3641 a(uniform) p 2212 3641 a(magnetic) p 2631 3641 a(\014eld) p 2844 3641 a(mo) m(v) m(es) p 3136 3641 a(along) p 3398 3641 a(a) p 3480 3641 a(circle) p 3737 3641 a(\(cy-) 0 3761 y(clotron) p 340 3761 a(motion\).) p 773 3761 a(The) p 983 3761 a(energy) p 1304 3761 a(of) p 1424 3761 a(an) p 1568 3761 a(electron) p 1946 3761 a(is) p 2053 3761 a(quan) m(tized) p 2507 3761 a(b) m(y) p 2651 3761 a(the) p 2828 3761 a(condition) p 3265 3761 a(that) p 3486 3761 a(the) p 3663 3761 a(phase) 0 3881 y(v) p 46 3881 a(ariation) p 417 3881 a(of) p 536 3881 a(the) p 712 3881 a(electron) p 1089 3881 a(w) m(a) m(v) m(e) p 1336 3881 a(in) p 1458 3881 a(one) p 1644 3881 a(cyclotron) p 2078 3881 a(rotation) p 2460 3881 a(is) p 2566 3881 a(2) p Fn(\031) p Fo 2714 3881 a(times) p 2983 3881 a(an) p 3127 3881 a(in) m(teger.) p 3511 3881 a(Th) m(us) p 3767 3881 a(the) 0 4002 y(energy) p 312 4002 a(of) p 423 4002 a(an) p 558 4002 a(electron) p 927 4002 a(tak) m(es) p 1177 4002 a(one) p 1355 4002 a(of) p 1466 4002 a(the) p 1635 4002 a(v) p 1681 4002 a(alues) p 1925 4002 a(in) p 2039 4002 a(the) p 2207 4002 a(Landau) p 2560 4002 a(lev) m(els.) 146 4122 y(If) p 248 4122 a(some) p 496 4122 a(solenoids) p 913 4122 a(are) p 1079 4122 a(con) m(tained) p 1524 4122 a(in) p 1642 4122 a(the) p 1814 4122 a(circle) p 2072 4122 a(of) p 2187 4122 a(the) p 2359 4122 a(cyclotron) p 2788 4122 a(motion,) p 3150 4122 a(then) p 3376 4122 a(the) p 3548 4122 a(phase) p 3823 4122 a(of) 0 4242 y(the) p 176 4242 a(electron) p 552 4242 a(w) m(a) m(v) m(e) p 798 4242 a(is) p 904 4242 a(shifted) p 1229 4242 a(b) m(y) p Fn 1372 4242 a(e=) p Fo(\026) p Fn 1465 4242 a(h) p Fo 1561 4242 a(times) p 1829 4242 a(the) p 2005 4242 a(magnetic) p 2430 4242 a(\015ux) p 2630 4242 a(of) p 2749 4242 a(solenoids) p 3169 4242 a(in) p 3291 4242 a(the) p 3466 4242 a(circle) p 3729 4242 a(\(the) 0 4363 y(Aharono) m(v-Bohm) p 726 4363 a(phase) p 991 4363 a(shift\).) p 1285 4363 a(Th) m(us) p 1525 4363 a(the) p 1687 4363 a(energy) p 1992 4363 a(of) p 2096 4363 a(the) p 2257 4363 a(electron) p 2619 4363 a(is) p 2711 4363 a(obliged) p 3040 4363 a(to) p 3153 4363 a(c) m(hange,) p 3497 4363 a(to) p 3609 4363 a(correct) 0 4483 y(the) p 158 4483 a(phase) p 420 4483 a(shift) p 630 4483 a(caused) p 936 4483 a(b) m(y) p 1061 4483 a(the) p 1220 4483 a(magnetic) p 1627 4483 a(\015ux) p 1810 4483 a(of) p 1911 4483 a(solenoids.) p 2359 4483 a(Hence) p 2639 4483 a(the) p 2798 4483 a(sp) s(ectrum) p 3214 4483 a(b) s(et) m(w) m(een) p 3581 4483 a(Landau) 0 4604 y(lev) m(els) p 261 4604 a(app) s(ears.) 146 4724 y(F) p 202 4724 a(or) p 325 4724 a(this) p 520 4724 a(reason,) p 856 4724 a(the) p 1028 4724 a(n) m(um) m(b) s(er) p 1387 4724 a(of) p 1502 4724 a(eigenstates) p 2001 4724 a(with) p 2227 4724 a(an) p 2366 4724 a(energy) p 2682 4724 a(b) s(et) m(w) m(een) p Fn 3063 4724 a(n) p Fo 3158 4724 a(th) p 3287 4724 a(and) p 3480 4724 a(\() p Fn(n) p Fo 3601 4724 a(+) p 3702 4724 a(1\)) p 3826 4724 a(st) 0 4844 y(Landau) p 351 4844 a(lev) m(el) p 570 4844 a(is) p 666 4844 a(roughly) p 1018 4844 a(estimated) p 1460 4844 a(b) m(y) p 1593 4844 a(the) p 1758 4844 a(p) s(ossible) p 2122 4844 a(n) m(um) m(b) s(er) p 2474 4844 a(of) p 2582 4844 a(electrons) p 2986 4844 a(with) p 3206 4844 a(the) p Fn 3371 4844 a(n) p Fo 3459 4844 a(th) p 3581 4844 a(Landau) 0 4965 y(lev) m(el) p 230 4965 a(energy) p 500 4965 a(,) p 570 4965 a(in) p 691 4965 a(the) p 867 4965 a(circle) p 1129 4965 a(of) p 1248 4965 a(the) p 1424 4965 a(Larmor) p 1780 4965 a(radius) p 2081 4965 a(cen) m(tered) p 2476 4965 a(at) p 2603 4965 a(the) p 2778 4965 a(p) s(osition) p 3158 4965 a(of) p 3276 4965 a(solenoid.) p 3712 4965 a(This) 1927 5214 y(4) p eop %%Page: 5 5 5 4 bop Fo 0 631 a(n) m(um) m(b) s(er) p 350 631 a(is) p 443 631 a(calculated) p 898 631 a(as) p 1013 631 a(follo) m(ws.) p 1370 631 a(If) p 1462 631 a(w) m(e) p 1601 631 a(normalize) p 2040 631 a(ph) m(ysical) p 2409 631 a(constan) m(ts) p 2836 631 a(as) p 2951 631 a(the) p 3114 631 a(mass) p Fn 3348 631 a(m) p Fo 3461 631 a(=) p 3565 631 a(1) p Fn(=) p Fo(2,) p 3767 631 a(the) 0 751 y(Planc) m(k) p 313 751 a(constan) m(t) p 698 751 a(\(divided) p 1072 751 a(b) m(y) p 1199 751 a(2) p Fn(\031) p Fo 1307 751 a(\)) p 1369 751 a(\026) p Fn 1368 751 a(h) p Fo 1452 751 a(=) p 1555 751 a(1) p 1628 751 a(and) p 1809 751 a(the) p 1969 751 a(c) m(harge) p 2267 751 a(of) p 2370 751 a(an) p 2497 751 a(electron) p Fn 2857 751 a(e) p Fo 2930 751 a(=) p 3034 751 a(1,) p 3135 751 a(then) p 3349 751 a(the) p 3509 751 a(cyclotron) 0 891 y(radius) p Fn 288 891 a(r) p Fo 362 891 a(of) p 469 891 a(an) p 599 891 a(electron) p 963 891 a(with) p Fn 1180 891 a(n) p Fo 1266 891 a(th) p 1385 891 a(Landau) p 1734 891 a(lev) m(el) p 1951 891 a(energy) p 2258 891 a(\(2) p Fn(n) p Fk 2415 891 a(\000) p Fo 2504 891 a(1\)) p Fn(B) p Fo 2697 891 a(equals) p 2988 891 a(to) p Fi 3103 791 a(q) p 3186 791 482 4 v Fo 3186 891 a(\(2) p Fn(n) p Fk 3353 891 a(\000) p Fo 3452 891 a(1\)) p Fn(=B) p Fo 3667 891 a(.) p 3736 891 a(It) p 3836 891 a(is) 0 1020 y(kno) m(wn) p 309 1020 a(that) p 521 1020 a(the) p 690 1020 a(densit) m(y) p 1027 1020 a(of) p 1138 1020 a(states) p 1416 1020 a(\(the) p 1622 1020 a(n) m(um) m(b) s(er) p 1978 1020 a(of) p 2089 1020 a(eigenstates) p 2584 1020 a(p) s(er) p 2755 1020 a(unit) p 2962 1020 a(area\)) p 3211 1020 a(for) p 3361 1020 a(eac) m(h) p 3581 1020 a(Landau) 0 1141 y(lev) m(el) p 227 1141 a(is) p Fn 330 1141 a(B) p 409 1141 a(=) p Fo(2) p Fn(\031) p Fo 603 1141 a(\(see) p 804 1141 a([Nak,) p 1070 1141 a(Prop) s(osition) p 1600 1141 a(15]\).) p 1848 1141 a(Th) m(us,) p 2128 1141 a(the) p 2301 1141 a(n) m(um) m(b) s(er) p 2661 1141 a(of) p 2777 1141 a(p) s(ossible) p 3149 1141 a(eigenstates) p 3648 1141 a(in) p 3767 1141 a(the) 0 1261 y(circle) p 255 1261 a(is) p Fn 1550 1405 a(\031) t(r) p Fl 1656 1364 a(2) p Fk 1717 1405 a(\002) p Fn 1841 1337 a(B) p 1827 1382 108 4 v Fo 1827 1473 a(2) p Fn(\031) p Fo 1972 1405 a(=) p Fn 2076 1405 a(n) p Fk 2156 1405 a(\000) p Fo 2266 1337 a(1) p 2266 1382 49 4 v 2266 1473 a(2) p Fn 2325 1405 a(:) p Fo 0 1593 a(The) p 196 1593 a(di\013erence) p 627 1593 a(b) s(et) m(w) m(een) p 999 1593 a(this) p 1184 1593 a(estimate) p 1570 1593 a(and) p 1755 1593 a(the) p 1918 1593 a(rigorous) p 2288 1593 a(result) p 2554 1593 a(\(\(i\)) p 2723 1593 a(of) p 2829 1593 a(Theorem) p 3236 1593 a(1.1\)) p 3426 1593 a(is) p 3519 1593 a(only) p 3729 1593 a(1) p Fn(=) p Fo(2.) 146 1714 y(When) p Fn 437 1714 a(N) p Fk 564 1714 a(\025) p Fo 679 1714 a(2,) p 795 1714 a(Nam) m(bu) p 1143 1714 a(\([Nam]\)) p 1515 1714 a(giv) m(es) p 1760 1714 a(a) p 1847 1714 a(represen) m(tation) p 2491 1714 a(of) p 2608 1714 a(eigenfunctions) p 3252 1714 a(for) p 3407 1714 a(the) p 3581 1714 a(Landau) 0 1834 y(lev) m(els) p 250 1834 a(b) m(y) p 375 1834 a(the) p 533 1834 a(m) m(ultiple) p 905 1834 a(in) m(tegral) p 1249 1834 a(in) p 1353 1834 a(the) p 1511 1834 a(complex) p 1882 1834 a(plane.) p 2177 1834 a(But) p 2360 1834 a(no) p 2485 1834 a(information) p 3004 1834 a(ab) s(out) p 3272 1834 a(the) p 3430 1834 a(eigen) m(v) p 3690 1834 a(alues) 0 1954 y(b) s(et) m(w) m(een) p 380 1954 a(the) p 552 1954 a(Landau) p 909 1954 a(lev) m(els) p 1173 1954 a(are) p 1339 1954 a(kno) m(wn.) p 1696 1954 a(Ho) m(w) m(ev) m(er,) p 2121 1954 a(the) p 2292 1954 a(ph) m(ysical) p 2670 1954 a(explanation) p 3202 1954 a(ab) s(o) m(v) m(e) p 3482 1954 a(giv) m(es) p 3725 1954 a(us) p 3853 1954 a(a) 0 2075 y(conjecture) p 474 2075 a(ab) s(out) p 758 2075 a(the) p 931 2075 a(n) m(um) m(b) s(er) p 1291 2075 a(of) p 1407 2075 a(eigen) m(v) p 1667 2075 a(alues) p 1916 2075 a(in) p 2035 2075 a(a) p 2122 2075 a(gap) p 2311 2075 a(of) p 2427 2075 a(Landau) p 2785 2075 a(lev) m(els,) p 3080 2075 a(when) p Fn 3339 2075 a(N) p Fk 3464 2075 a(\025) p Fo 3578 2075 a(2.) p 3712 2075 a(This) 0 2195 y(n) m(um) m(b) s(er) p 365 2195 a(is) p 474 2195 a(roughly) p 840 2195 a(estimated) p 1295 2195 a(b) m(y) p 1441 2195 a(the) p 1619 2195 a(n) m(um) m(b) s (er) p 1985 2195 a(of) p 2106 2195 a(eigenstates) p 2611 2195 a(with) p 2844 2195 a(the) p Fn 3022 2195 a(n) p Fo 3124 2195 a(th) p 3259 2195 a(Landau) p 3623 2195 a(energy) 0 2315 y(in) p 122 2315 a(the) p 298 2315 a(union) p 577 2315 a(set,) p 767 2315 a(with) p 997 2315 a(resp) s(ect) p 1339 2315 a(to) p Fn 1466 2315 a(j) p Fo 1554 2315 a(=) p 1671 2315 a(1) p Fn(;) p 1764 2315 a(:) p 1808 2315 a(:) p 1852 2315 a(:) p 1895 2315 a(;) p 1939 2315 a(N) p Fo 2027 2315 a(,) p 2097 2315 a(of) p 2216 2315 a(the) p 2392 2315 a(disks) p 2643 2315 a(of) p 2762 2315 a(Larmor) p 3118 2315 a(radius) p 3420 2315 a(cen) m(tered) p 3815 2315 a(at) p Fn 0 2436 a(z) p Fm 45 2451 a(j) p Fo 82 2436 a(.) p 170 2436 a(Eac) m(h) p 418 2436 a(disk) p 628 2436 a(con) m(tains) p Fn 1017 2436 a(n) p Fo 1113 2436 a(eigenstates) p 1613 2436 a(with) p Fn 1842 2436 a(n) p Fo 1938 2436 a(th) p 2069 2436 a(Landau) p 2428 2436 a(energy) p 2698 2436 a(.) p 2787 2436 a(These) p 3076 2436 a(disks) p 3324 2436 a(ma) m(y) p 3541 2436 a(in) m(tersect) 0 2556 y(in) p 123 2556 a(general,) p 499 2556 a(but) p 687 2556 a(they) p 916 2556 a(are) p 1089 2556 a(disjoin) m(t) p 1446 2556 a(if) p 1545 2556 a(solenoids) p 1967 2556 a(are) p 2140 2556 a(far) p 2298 2556 a(from) p 2538 2556 a(eac) m(h) p 2768 2556 a(other.) p 3089 2556 a(Th) m(us) p 3346 2556 a(w) m(e) p 3499 2556 a(reac) m(h) p 3767 2556 a(the) 0 2677 y(follo) m(wing) p 412 2677 a(conjecture.) p Fq 146 2907 a(Conjecture) p Ff 770 2907 a(\(I\)) p 924 2907 a(The) p 1126 2907 a(numb) p 1353 2907 a(er) p 1477 2907 a(of) p 1594 2907 a(eigenvalues) p 2107 2907 a(b) p 2147 2907 a(etwe) p 2329 2907 a(en) p Fn 2466 2907 a(n) p Ff 2562 2907 a(th) p 2681 2907 a(and) p Fo 2873 2907 a(\() p Fn(n) p Fo 2994 2907 a(+) p 3094 2907 a(1\)) p Ff 3218 2907 a(st) p 3328 2907 a(L) p 3384 2907 a(andau) p 3678 2907 a(levels) 0 3027 y(is) p 105 3027 a(b) p 145 3027 a(ounde) p 392 3027 a(d) p 476 3027 a(by) p Fn 603 3027 a(nN) p Ff 749 3027 a(.) 146 3147 y(\(II\)) p 336 3147 a(If) p 439 3147 a(solenoids) p 858 3147 a(ar) p 944 3147 a(e) p 1024 3147 a(far) p 1180 3147 a(fr) p 1246 3147 a(om) p 1411 3147 a(e) p 1451 3147 a(ach) p 1631 3147 a(other) p 1884 3147 a(c) p 1924 3147 a(omp) p 2099 3147 a(ar) p 2185 3147 a(e) p 2225 3147 a(d) p 2309 3147 a(with) p 2521 3147 a(the) p 2684 3147 a(cyclotr) p 2964 3147 a(on) p 3104 3147 a(r) p 3140 3147 a(adius,) p 3427 3147 a(the) p 3589 3147 a(numb) p 3816 3147 a(er) 0 3268 y(of) p 115 3268 a(eigenvalues) p 625 3268 a(b) p 665 3268 a(etwe) p 847 3268 a(en) p Fn 981 3268 a(n) p Ff 1074 3268 a(th) p 1191 3268 a(and) p Fo 1381 3268 a(\() p Fn(n) p Fo 1499 3268 a(+) p 1597 3268 a(1\)) p Ff 1719 3268 a(st) p 1826 3268 a(L) p 1882 3268 a(andau) p 2173 3268 a(levels) p 2432 3268 a(e) p 2472 3268 a(quals) p 2719 3268 a(to) p Fn 2836 3268 a(nN) p Ff 2982 3268 a(.) p Fo 146 3451 a(Our) p 359 3451 a(aim) p 561 3451 a(in) p 687 3451 a(this) p 889 3451 a(pap) s(er) p 1175 3451 a(is) p 1285 3451 a(to) p 1417 3451 a(giv) m(e) p 1629 3451 a(an) p 1777 3451 a(answ) m(er) p 2112 3451 a(to) p 2244 3451 a(these) p 2506 3451 a(conjectures.) p 3087 3451 a(Our) p 3300 3451 a(answ) m(er) p 3635 3451 a(to) p 3767 3451 a(the) 0 3571 y(conjecture) p 469 3571 a(\(I\)) p 612 3571 a(is) p 710 3571 a(the) p 878 3571 a(follo) m(wing.) p Fq 146 3740 a(Theorem) p 621 3740 a(1.2) p Ff 813 3740 a(L) p 869 3740 a(et) p Fo 982 3740 a(1) p Fk 1058 3740 a(\024) p Fn 1163 3740 a(N) p 1279 3740 a(<) p Fk 1383 3740 a(1) p Ff(.) p 1557 3740 a(Then,) p 1841 3740 a(the) p 2003 3740 a(fol) p 2113 3740 a(lowing) p 2417 3740 a(holds.) 146 3861 y(\(i\)) p 301 3861 a(F) p 358 3861 a(or) p 493 3861 a(any) p 689 3861 a(self-adjoint) p 1205 3861 a(extension) p Fn 1646 3861 a(H) p Fm 1727 3876 a(N) p Ff 1839 3861 a(of) p Fn 1963 3861 a(L) p Fm 2029 3876 a(N) p Ff 2097 3861 a(,) p 2174 3861 a(we) p 2329 3861 a(have) p 2563 3861 a(that) p Fo 2772 3861 a(\(2) p Fn(n) p Fk 2947 3861 a(\000) p Fo 3054 3861 a(1\)) p Fn(B) p Ff 3264 3861 a(is) p 3379 3861 a(an) p 3528 3861 a(in\014nitely) 0 3981 y(de) p 90 3981 a(gener) p 316 3981 a(ate) p 438 3981 a(d) p 522 3981 a(eigenvalue) p 993 3981 a(for) p 1148 3981 a(every) p Fn 1406 3981 a(n) p Fo 1492 3981 a(=) p 1595 3981 a(1) p Fn(;) p Fo 1688 3981 a(2) p Fn(;) p Fo 1781 3981 a(3) p Fn(;) p 1874 3981 a(:) p 1918 3981 a(:) p 1962 3981 a(:) p Ff 1987 3981 a(.) 146 4101 y(\(ii\)) p 321 4101 a(F) p 378 4101 a(or) p 503 4101 a(the) p 664 4101 a(standar) p 977 4101 a(d) p 1062 4101 a(A) n(har) p 1268 4101 a(onov-Bohm) p 1784 4101 a(Hamiltonian) p Fn 2347 4101 a(H) p Fm 2436 4065 a(AB) 2428 4126 y(N) p Ff 2549 4101 a(,) p 2614 4101 a(we) p 2759 4101 a(have) p Fo 1034 4282 a(dim) p 1213 4282 a(Ran) p Fn 1404 4282 a(P) p Fl 1467 4297 a(\() p Fg(\0001) p Fm(;B) p Fl 1696 4297 a(\)) p Fo 1728 4282 a(\() p Fn(H) p Fm 1855 4240 a(AB) 1847 4306 y(N) p Fo 1968 4282 a(\)) p 2090 4282 a(=) p 2249 4282 a(0) p Fn(;) p Fo 656 4427 a(dim) p 836 4427 a(Ran) p Fn 1027 4427 a(P) p Fl 1090 4442 a(\(\(2) p Fm(n) p Fg(\000) p Fl(1\)) p Fm(B) s(;) p Fl(\(2) p Fm(n) p Fl(+1\)) p Fm(B) p Fl 1693 4442 a(\)) p Fo 1728 4427 a(\() p Fn(H) p Fm 1855 4386 a(AB) 1847 4452 y(N) p Fo 1968 4427 a(\)) p Fk 2089 4427 a(\024) p Fn 2249 4427 a(nN) p 2390 4427 a(;) p Fo 2469 4427 a(for) p Fn 2621 4427 a(n) p Fo 2706 4427 a(=) p 2810 4427 a(1) p Fn(;) p Fo 2903 4427 a(2) p Fn(;) p Fo 2996 4427 a(3) p Fn(;) p 3089 4427 a(:) p 3133 4427 a(:) p 3177 4427 a(:) p 3219 4427 a(:) p Fo 3701 4427 a(\(1.9\)) p Ff 146 4607 a(\(iii\)) p 351 4607 a(F) p 408 4607 a(or) p 532 4607 a(any) p 719 4607 a(self-adjoint) p 1225 4607 a(extension) p Fn 1656 4607 a(H) p Fm 1737 4622 a(N) p Ff 1839 4607 a(of) p Fn 1953 4607 a(L) p Fm 2019 4622 a(N) p Ff 2087 4607 a(,) p 2152 4607 a(we) p 2296 4607 a(have) p Fo 938 4787 a(dim) p 1118 4787 a(Ran) p Fn 1309 4787 a(P) p Fl 1372 4803 a(\() p Fg(\0001) p Fm(;B) p Fl 1601 4803 a(\)) p Fo 1632 4787 a(\() p Fn(H) p Fm 1751 4802 a(N) p Fo 1818 4787 a(\)) p Fk 1939 4787 a(\024) p Fo 2100 4787 a(2) p Fn(N) p 2232 4787 a(;) p Fo 3653 4787 a(\(1.10\)) 561 4933 y(dim) p 740 4933 a(Ran) p Fn 932 4933 a(P) p Fl 995 4948 a(\(\(2) p Fm(n) p Fg(\000) p Fl(1\)) p Fm(B) s(;) p Fl(\(2) p Fm(n) p Fl(+1\)) p Fm(B) p Fl 1598 4948 a(\)) p Fo 1632 4933 a(\() p Fn(H) p Fm 1751 4948 a(N) p Fo 1818 4933 a(\)) p Fk 1939 4933 a(\024) p Fo 2100 4933 a(\() p Fn(n) p Fo 2218 4933 a(+) p 2316 4933 a(1\)) p Fn(N) p 2486 4933 a(;) p Fo 2565 4933 a(for) p Fn 2716 4933 a(n) p Fo 2802 4933 a(=) p 2905 4933 a(1) p Fn(;) p Fo 2998 4933 a(2) p Fn(;) p Fo 3091 4933 a(3) p Fn(;) p 3184 4933 a(:) p 3228 4933 a(:) p 3272 4933 a(:) p 3314 4933 a(:) p Fo 3653 4933 a(\(1.11\)) 1927 5214 y(5) p eop %%Page: 6 6 6 5 bop Fo 0 631 a(In) p 122 631 a(the) p 290 631 a(case) p 496 631 a(of) p Fn 607 631 a(N) p Fo 724 631 a(=) p 827 631 a(1,) p 936 631 a(our) p 1109 631 a(result) p 1381 631 a(and) p 1570 631 a(\(1.8\)) p 1803 631 a(imply) p 2077 631 a(that) p Fn 956 827 a(n) p Fk 1037 827 a(\000) p Fo 1136 827 a(1) p Fk 1213 827 a(\024) p Fo 1318 827 a(dim) p 1497 827 a(Ran) p Fn 1689 827 a(P) p Fl 1752 843 a(\(\(2) p Fm(n) p Fg(\000) p Fl(1\)) p Fm(B) s(;) p Fl(\(2) p Fm(n) p Fl(+1\)) p Fm(B) p Fl 2355 843 a(\)) p Fo 2389 827 a(\() p Fn(H) p Fl 2508 842 a(1) p Fo 2548 827 a(\)) p Fk 2613 827 a(\024) p Fn 2718 827 a(n) p Fo 2799 827 a(+) p 2897 827 a(1) p 3653 827 a(\(1.12\)) 0 1024 y(for) p Fn 142 1024 a(n) p Fo 228 1024 a(=) p 331 1024 a(1) p Fn(;) p Fo 424 1024 a(2) p Fn(;) p 517 1024 a(:) p 561 1024 a(:) p 605 1024 a(:) p Fo 631 1024 a(.) p 699 1024 a(The) p 892 1024 a(upp) s(er) p 1164 1024 a(b) s(ound) p 1457 1024 a(of) p 1561 1024 a(\(1.12\)) p 1836 1024 a(is) p 1926 1024 a(sharp) s(er) p 2269 1024 a(than) p 2490 1024 a(that) p 2694 1024 a(of) p 2797 1024 a(\(1.8\)) p 3023 1024 a(\(ho) m(w) m(ev) m(er,) p 3456 1024 a([Ex-St-Vy) p 3874 1024 a(,) 0 1144 y(Fig) p 179 1144 a(1,2]) p 371 1144 a(seems) p 656 1144 a(to) p 783 1144 a(indicate) p 1159 1144 a(that) p 1377 1144 a(there) p 1634 1144 a(are) p 1804 1144 a(at) p 1931 1144 a(most) p 2177 1144 a(t) m(w) m(o) p 2369 1144 a(eigen) m(v) p 2629 1144 a(alues) p 2881 1144 a(of) p Fn 2999 1144 a(H) p Fl 3080 1159 a(1) p Fk 3120 1144 a(j) p Fg 3148 1159 a(H) p Fe 3208 1167 a(c) p Fo 3284 1144 a(in) p 3405 1144 a(eac) m(h) p 3632 1144 a(gap) p 3823 1144 a(of) 0 1264 y(Landau) p 354 1264 a(lev) m(els\).) 146 1385 y(Next,) p 412 1385 a(w) m(e) p 555 1385 a(shall) p 783 1385 a(exhibit) p 1111 1385 a(our) p 1284 1385 a(answ) m(er) p 1607 1385 a(to) p 1726 1385 a(the) p 1894 1385 a(conjecture) p 2362 1385 a(\(I) s(I\).) p 2571 1385 a(In) p 2693 1385 a(this) p 2883 1385 a(pap) s(er,) p 3183 1385 a(w) m(e) p 3327 1385 a(shall) p 3555 1385 a(consider) 0 1505 y(the) p 161 1505 a(sp) s(ecial) p 472 1505 a(case) p 672 1505 a(where) p 947 1505 a(the) p 1108 1505 a(ph) m(ysical) p 1476 1505 a(situations) p 1914 1505 a(around) p 2238 1505 a(ev) m(ery) p Fn 2489 1505 a(z) p Fm 2534 1520 a(j) p Fo 2597 1505 a(are) p 2752 1505 a(the) p 2914 1505 a(same.) p 3194 1505 a(T) p 3256 1505 a(o) p 3331 1505 a(represen) m(t) p 3744 1505 a(this) 0 1626 y(situation) p 397 1626 a(rigorously) p 811 1626 a(,) p 861 1626 a(w) m(e) p 994 1626 a(shall) p 1212 1626 a(prepare) p 1554 1626 a(an) p 1679 1626 a(op) s(erator) p 2062 1626 a(whic) m(h) p 2331 1626 a(in) m(tert) m(wines) p 2820 1626 a(t) m(w) m(o) p 2994 1626 a(magnetic) p 3401 1626 a(Sc) m(hr\177) p 3587 1626 a(odinger) 0 1746 y(op) s(erators.) p Fq 146 1929 a(De\014nition) p 669 1929 a(1.1) p Ff 861 1929 a(L) p 917 1929 a(et) p Fn 1032 1929 a(w) p Fk 1137 1929 a(2) p Fj 1236 1929 a(R) p Fl 1323 1887 a(2) p Ff 1363 1929 a(.) p 1445 1929 a(L) p 1501 1929 a(et) p Fn 1616 1929 a(U) p Ff 1730 1929 a(b) p 1770 1929 a(e) p 1853 1929 a(a) p 1940 1929 a(simply) p 2249 1929 a(c) p 2289 1929 a(onne) p 2489 1929 a(cte) p 2606 1929 a(d) p 2692 1929 a(op) p 2787 1929 a(en) p 2924 1929 a(set,) p 3109 1929 a(and) p Fn 3301 1929 a(V) p Fo 3412 1929 a(=) p Fn 3521 1929 a(U) p Fo 3621 1929 a(+) p Fn 3721 1929 a(w) p Fo 3826 1929 a(=) p Fk 0 2049 a(f) p Fn(z) p Fo 115 2049 a(+) p Fn 206 2049 a(w) p Fo 279 2049 a(;) p Fn 323 2049 a(z) p Fk 400 2049 a(2) p Fn 494 2049 a(U) p Fk 570 2049 a(g) p Ff(.) p 694 2049 a(L) p 750 2049 a(et) p Fn 859 2049 a(S) p Ff 957 2049 a(b) p 997 2049 a(e) p 1073 2049 a(an) p 1210 2049 a(at) p 1324 2049 a(most) p 1557 2049 a(c) p 1597 2049 a(ountable) p 1982 2049 a(subset) p 2268 2049 a(of) p Fn 2380 2049 a(U) p Ff 2488 2049 a(with) p 2697 2049 a(no) p 2833 2049 a(ac) p 2923 2049 a(cumulation) p 3425 2049 a(p) p 3470 2049 a(oints) p 3709 2049 a(in) p Fn 3825 2049 a(U) p Ff 0 2170 a(and) p Fn 188 2170 a(T) p Fo 286 2170 a(=) p Fn 390 2170 a(S) p Fo 475 2170 a(+) p Fn 571 2170 a(w) p Ff 644 2170 a(.) p 717 2170 a(L) p 773 2170 a(et) p Fj 884 2170 a(a) p Fk 975 2170 a(2) p Fn 1069 2170 a(C) p Fg 1146 2134 a(1) p Fo 1221 2170 a(\() p Fn(U) p Fk 1355 2170 a(n) p Fn 1424 2170 a(S) p Fo 1490 2170 a(;) p Fj 1534 2170 a(R) p Fl 1621 2128 a(2) p Fo 1660 2170 a(\)) p Fk 1718 2170 a(\\) p Fn 1803 2170 a(L) p Fl 1869 2134 a(1) p Fm 1869 2194 a(l) q(oc) p Fo 1960 2170 a(\() p Fn(U) p Fo 2074 2170 a(;) p Fj 2118 2170 a(R) p Fl 2206 2128 a(2) p Fo 2246 2170 a(\)) p Ff 2317 2170 a(and) p Fj 2505 2170 a(b) p Fk 2585 2170 a(2) p Fn 2679 2170 a(C) p Fg 2756 2134 a(1) p Fo 2831 2170 a(\() p Fn(V) p Fk 2966 2170 a(n) p Fn 3036 2170 a(T) p Fo 3107 2170 a(;) p Fj 3151 2170 a(R) p Fl 3237 2128 a(2) p Fo 3277 2170 a(\)) p Fk 3334 2170 a(\\) p Fn 3420 2170 a(L) p Fl 3486 2134 a(1) p Fm 3486 2194 a(l) q(oc) p Fo 3577 2170 a(\() p Fn(V) p Fo 3694 2170 a(;) p Fj 3738 2170 a(R) p Fl 3825 2128 a(2) p Fo 3864 2170 a(\)) p Ff 0 2290 a(b) p 40 2290 a(e) p 120 2290 a(two) p 301 2290 a(ve) p 386 2290 a(ctor) p 589 2290 a(p) p 634 2290 a(otentials) p 1027 2290 a(satisfying) p Fo 1465 2486 a(rot) p Fj 1606 2486 a(a) p Fo(\() p Fn(z) p Fo 1756 2486 a(\)) p 1822 2486 a(=) p 1926 2486 a(rot) p Fj 2067 2486 a(b) p Fo(\() p Fn(z) p Fo 2229 2486 a(+) p Fn 2327 2486 a(w) p Fo 2400 2486 a(\)) p Ff 0 2683 a(in) p Fk 127 2683 a(D) p Fg 207 2647 a(0) p Fo 230 2683 a(\() p Fn(U) p Fo 344 2683 a(\)) p Ff(.) p 480 2683 a(Then,) p 774 2683 a(ther) p 937 2683 a(e) p 1024 2683 a(exists) p 1298 2683 a(an) p 1446 2683 a(op) p 1541 2683 a(er) p 1622 2683 a(ator) p Fn 1837 2683 a(t) p Fg 1872 2698 a(\000) p Fm(w) p Ff 2026 2683 a(fr) p 2092 2683 a(om) p Fk 2264 2683 a(D) p Fg 2344 2647 a(0) p Fo 2367 2683 a(\() p Fn(V) p Fk 2511 2683 a(n) p Fn 2589 2683 a(T) p Fo 2660 2683 a(\)) p Ff 2740 2683 a(to) p Fk 2865 2683 a(D) p Fg 2945 2647 a(0) p Fo 2968 2683 a(\() p Fn(U) p Fk 3110 2683 a(n) p Fn 3188 2683 a(S) p Fo 3254 2683 a(\)) p Ff 3334 2683 a(satisfying) p 3775 2683 a(the) 0 2803 y(fol) p 110 2803 a(lowing) p 413 2803 a(\(i\)) p 558 2803 a(and) p 747 2803 a(\(ii\):) 146 2924 y(\(i\)) p 290 2924 a(Ther) p 491 2924 a(e) p 569 2924 a(exists) p 834 2924 a(a) p 918 2924 a(c) p 958 2924 a(omplex-value) p 1500 2924 a(d) p 1582 2924 a(smo) p 1747 2924 a(oth) p 1912 2924 a(function) p Fo 2294 2924 a(\010\() p Fn(z) p Fo 2451 2924 a(\)) p Fk 2518 2924 a(2) p Fn 2612 2924 a(C) p Fg 2689 2887 a(1) p Fo 2764 2924 a(\() p Fn(U) p Fk 2898 2924 a(n) p Fn 2968 2924 a(S) p Fo 3034 2924 a(\)) p Ff 3105 2924 a(with) p Fk 3316 2924 a(j) p Fo(\010\() p Fn(z) p Fo 3501 2924 a(\)) p Fk(j) p Fo 3595 2924 a(=) p 3699 2924 a(1) p Ff 3781 2924 a(for) 0 3044 y(every) p Fn 258 3044 a(z) p Fk 335 3044 a(2) p Fn 429 3044 a(U) p Fk 528 3044 a(n) p Fn 600 3044 a(S) p Ff 666 3044 a(,) p 731 3044 a(such) p 952 3044 a(that) p Fn 1091 3240 a(t) p Fg 1126 3255 a(\000) p Fm(w) p Fn 1238 3240 a(v) p Fo 1289 3240 a(\() p Fn(z) p Fo 1376 3240 a(\)) p 1442 3240 a(=) p 1545 3240 a(\010\() p Fn(z) p Fo 1702 3240 a(\)) p Fn(v) p Fo 1791 3240 a(\() p Fn(z) p Fo 1902 3240 a(+) p Fn 2000 3240 a(w) p Fo 2073 3240 a(\)) p Fn(;) p 2189 3240 a(v) p Fk 2267 3240 a(2) p 2362 3240 a(D) p Fg 2442 3199 a(0) p Fo 2464 3240 a(\() p Fn(V) p Fk 2603 3240 a(n) p Fn 2675 3240 a(T) p Fo 2746 3240 a(\)) p Fn(:) p Ff 146 3437 a(\(ii\)) p 321 3437 a(The) p 520 3437 a(fol) p 630 3437 a(lowing) p 933 3437 a(distributional) p 1530 3437 a(e) p 1570 3437 a(quality) p 1886 3437 a(holds:) p Fq 988 3633 a(p) p Fo(\() p Fj(a) p Fo(\)) p Fn(t) p Fg 1224 3648 a(\000) p Fm(w) p Fn 1336 3633 a(v) p Fo 1414 3633 a(=) p Fn 1518 3633 a(t) p Fg 1553 3648 a(\000) p Fm(w) p Fq 1665 3633 a(p) p Fo(\() p Fj(b) p Fo(\)) p Fn(v) t(;) p Fk 1984 3633 a(L) p Fo(\() p Fj(a) p Fo(\)) p Fn(t) p Fg 2227 3648 a(\000) p Fm(w) p Fn 2338 3633 a(v) p Fo 2417 3633 a(=) p Fn 2520 3633 a(t) p Fg 2555 3648 a(\000) p Fm(w) p Fk 2667 3633 a(L) p Fo(\() p Fj(b) p Fo 2825 3633 a(\)) p Fn(v) p Fo 3653 3633 a(\(1.13\)) p Ff 0 3829 a(for) p Fn 156 3829 a(v) p Fk 234 3829 a(2) p 328 3829 a(D) p Fg 408 3793 a(0) p Fo 431 3829 a(\() p Fn(V) p Fk 570 3829 a(n) p Fn 642 3829 a(T) p Fo 713 3829 a(\)) p Ff(,) p 815 3829 a(wher) p 1011 3829 a(e) p Fq 1176 4070 a(p) p Fo(\() p Fj(a) p Fo 1340 4070 a(\)) p 1405 4070 a(=) 1519 4003 y(1) p 1519 4047 49 4 v Fn 1527 4138 a(i) p Fk 1578 4070 a(r) p Fo 1683 4070 a(+) p Fj 1781 4070 a(a) p Fn(;) p Fq 2037 4070 a(p) p Fo(\() p Fj(b) p Fo(\)) p 2255 4070 a(=) 2368 4003 y(1) p 2368 4047 V Fn 2376 4138 a(i) p Fk 2427 4070 a(r) p Fo 2532 4070 a(+) p Fj 2630 4070 a(b) p Fn(;) p Fk 1264 4255 a(L) p Fo(\() p Fj(a) p Fo(\)) p 1500 4255 a(=) p Fq 1603 4255 a(p) p Fo(\() p Fj(a) p Fo(\)) p Fl 1804 4214 a(2) p Fn 1844 4255 a(;) p Fk 2037 4255 a(L) p Fo(\() p Fj(b) p Fo 2195 4255 a(\)) p 2261 4255 a(=) p Fq 2365 4255 a(p) p Fo(\() p Fj(a) p Fo(\)) p Fl 2566 4214 a(2) p Fn 2605 4255 a(:) p Ff 0 4452 a(We) p 181 4452 a(c) p 221 4452 a(al) p 301 4452 a(l) p 363 4452 a(the) p 529 4452 a(op) p 624 4452 a(er) p 705 4452 a(ator) p Fn 916 4452 a(t) p Fg 951 4467 a(\000) p Fm(w) p Ff 1101 4452 a(the) p 1267 4452 a(magnetic) p 1686 4452 a(tr) p 1754 4452 a(anslation) p 2179 4452 a(op) p 2274 4452 a(er) p 2355 4452 a(ator) p 2566 4452 a(fr) p 2632 4452 a(om) p Fn 2800 4452 a(V) p Ff 2917 4452 a(to) p Fn 3038 4452 a(U) p Ff 3153 4452 a(intertwining) p Fk 3706 4452 a(L) p Fo(\() p Fj(b) p Fo 3864 4452 a(\)) p Ff 0 4572 a(with) p Fk 212 4572 a(L) p Fo(\() p Fj(a) p Fo 381 4572 a(\)) p Ff(.) p 494 4572 a(We) p 671 4572 a(denote) p 982 4572 a(the) p 1144 4572 a(inverse) p 1479 4572 a(op) p 1574 4572 a(er) p 1655 4572 a(ator) p 1863 4572 a(of) p Fn 1977 4572 a(t) p Fg 2012 4587 a(\000) p Fm(w) p Ff 2159 4572 a(by) p Fn 2286 4572 a(t) p Fm 2321 4587 a(w) p Ff 2378 4572 a(,) p 2443 4572 a(that) p 2642 4572 a(is,) p Fn 1366 4768 a(t) p Fm 1401 4783 a(w) p Fn 1458 4768 a(u) p Fo(\() p Fn(z) p Fo 1601 4768 a(\)) p 1667 4768 a(=) p 1770 4768 a(\010\() p Fn(z) p Fk 1951 4768 a(\000) p Fn 2050 4768 a(w) p Fo 2123 4768 a(\)) p Fn(u) p Fo(\() p Fn(z) p Fk 2326 4768 a(\000) p Fn 2426 4768 a(w) p Fo 2499 4768 a(\)) p Ff 0 4965 a(for) p Fn 156 4965 a(u) p Fk 239 4965 a(2) p 333 4965 a(D) p Fg 413 4929 a(0) p Fo 436 4965 a(\() p Fn(U) p Fk 573 4965 a(n) p Fn 645 4965 a(S) p Fo 711 4965 a(\)) p Ff(.) p Fo 1927 5214 a(6) p eop %%Page: 7 7 7 6 bop Fo 0 631 a(W) p 92 631 a(e) p 171 631 a(call) p 353 631 a(the) p 524 631 a(equalit) m(y) p 898 631 a(\(1.13\)) p 1183 631 a(the) p 1354 631 a(in) m(tert) m(wining) p 1904 631 a(prop) s(ert) m(y) p 2305 631 a(of) p Fn 2420 631 a(t) p Fg 2455 646 a(\000) p Fm(w) p Fo 2566 631 a(.) p 2646 631 a(The) p 2849 631 a(existence) p 3267 631 a(of) p 3381 631 a(the) p 3552 631 a(function) 0 751 y(\010) p 103 751 a(will) p 287 751 a(b) s(e) p 420 751 a(pro) m(v) m(ed) p 737 751 a(in) p 851 751 a(Prop) s(osition) p 1376 751 a(2.4) p 1533 751 a(b) s(elo) m(w.) p Fq 146 967 a(De\014nition) p 669 967 a(1.2) p Ff 861 967 a(L) p 917 967 a(et) p Fn 1030 967 a(H) p Fm 1111 982 a(N) p Ff 1214 967 a(b) p 1254 967 a(e) p 1334 967 a(a) p 1419 967 a(self-adjoint) p 1925 967 a(extension) p 2357 967 a(of) p Fn 2472 967 a(L) p Fm 2538 982 a(N) p Ff 2606 967 a(.) p 2682 967 a(We) p 2860 967 a(say) p 3032 967 a(the) p 3195 967 a(op) p 3290 967 a(er) p 3371 967 a(ator) p Fn 3579 967 a(H) p Fm 3660 982 a(N) p Ff 3763 967 a(has) 0 1088 y(the) p 162 1088 a(same) p 411 1088 a(b) p 451 1088 a(oundary) p 831 1088 a(c) p 871 1088 a(ondition) p 1257 1088 a(at) p 1374 1088 a(every) p Fn 1631 1088 a(z) p Fm 1676 1103 a(j) p Ff 1713 1088 a(,) p 1778 1088 a(if) p 1873 1088 a(the) p 2035 1088 a(fol) p 2145 1088 a(lowing) p 2448 1088 a(two) p 2630 1088 a(c) p 2670 1088 a(onditions) p 3096 1088 a(hold:) 146 1208 y(\(i\)) p 291 1208 a(Ther) p 492 1208 a(e) p 571 1208 a(exists) p 837 1208 a(a) p 922 1208 a(c) p 962 1208 a(onstant) p Fn 1311 1208 a(\013) p Ff 1408 1208 a(with) p Fo 1620 1208 a(0) p Fn 1696 1208 a(<) p 1800 1208 a(\013) p 1890 1208 a(<) p Fo 1994 1208 a(1) p Ff 2078 1208 a(such) p 2299 1208 a(that) p Fn 2498 1208 a(\013) p Fm 2560 1223 a(j) p Fo 2625 1208 a(=) p Fn 2728 1208 a(\013) p Ff 2826 1208 a(for) p 2981 1208 a(every) p Fn 3239 1208 a(j) p Fo 3313 1208 a(=) p 3416 1208 a(1) p Fn(;) p 3509 1208 a(:) p 3553 1208 a(:) p 3597 1208 a(:) p 3640 1208 a(;) p 3684 1208 a(N) p Ff 3772 1208 a(.) 146 1329 y(\(ii\)) p 333 1329 a(L) p 389 1329 a(et) p Fn 513 1329 a(t) p Fg 548 1344 a(\000) p Fm(z) p Fe 636 1354 a(j) p Ff 720 1329 a(b) p 760 1329 a(e) p 852 1329 a(the) p 1026 1329 a(magnetic) p 1454 1329 a(tr) p 1522 1329 a(anslation) p 1956 1329 a(op) p 2051 1329 a(er) p 2132 1329 a(ator) p 2352 1329 a(fr) p 2418 1329 a(om) p Fk 2594 1329 a(fj) p Fn(z) p Fk 2753 1329 a(\000) p Fn 2861 1329 a(z) p Fm 2906 1344 a(j) p Fk 2943 1329 a(j) p Fn 3021 1329 a(<) p Fm 3157 1289 a(R) p 3157 1305 54 4 v Fl 3166 1363 a(2) p Fk 3221 1329 a(g) p Ff 3318 1329 a(to) p Fk 3447 1329 a(fj) p Fn(z) p Fk 3574 1329 a(j) p Fn 3652 1329 a(<) p Fm 3789 1289 a(R) p 3789 1305 V Fl 3798 1363 a(2) p Fk 3852 1329 a(g) p Ff 0 1449 a(intertwining) p Fk 545 1449 a(L) p Fm 614 1464 a(N) p Ff 712 1449 a(with) p Fk 920 1449 a(L) p Fm 989 1395 a(\013) p Fe 1034 1405 a(j) p Fl 989 1471 a(1) p Ff 1071 1449 a(.) p 1144 1449 a(L) p 1200 1449 a(et) p Fn 1308 1449 a(\037) p Fk 1397 1449 a(2) p Fn 1491 1449 a(C) p Fg 1568 1413 a(1) p Fl 1561 1474 a(0) p Fo 1643 1449 a(\() p Fj(R) p Fl 1768 1407 a(2) p Fo 1808 1449 a(\)) p Ff 1876 1449 a(b) p 1916 1449 a(e) p 1992 1449 a(a) p 2073 1449 a(function) p 2452 1449 a(satisfying) p Fo 2882 1449 a(0) p Fk 2958 1449 a(\024) p Fn 3063 1449 a(\037) p Fk 3152 1449 a(\024) p Fo 3257 1449 a(1) p Ff 3337 1449 a(on) p Fj 3473 1449 a(R) p Fl 3560 1407 a(2) p Ff 3599 1449 a(,) p Fn 3661 1449 a(\037) p Fo 3750 1449 a(=) p 3853 1449 a(0) p Ff 0 1569 a(in) p Fk 115 1569 a(j) p Fn(z) p Fk 192 1569 a(j) p Fn 248 1569 a(>) p Fm 361 1530 a(R) p 361 1546 V Fl 370 1604 a(2) p Ff 455 1569 a(and) p Fn 640 1569 a(\037) p Fo 728 1569 a(=) p 832 1569 a(1) p Ff 911 1569 a(in) p Fk 1026 1569 a(j) p Fn(z) p Fk 1103 1569 a(j) p Fn 1159 1569 a(<) p Fm 1272 1530 a(R) p 1272 1546 V Fl 1281 1604 a(3) p Ff 1336 1569 a(.) p 1409 1569 a(Put) p Fn 1590 1569 a(\037) p Fm 1651 1584 a(j) p Fo 1688 1569 a(\() p Fn(z) p Fo 1775 1569 a(\)) p 1841 1569 a(=) p Fn 1944 1569 a(\037) p Fo(\() p Fn(z) p Fk 2104 1569 a(\000) p Fn 2193 1569 a(z) p Fm 2238 1584 a(j) p Fo 2276 1569 a(\)) p Ff(.) p 2387 1569 a(Ther) p 2588 1569 a(e) p 2663 1569 a(exists) p 2925 1569 a(a) p 3005 1569 a(self-adjoint) p 3506 1569 a(extension) p Fn 0 1690 a(H) p Fl 81 1705 a(1) p Ff 155 1690 a(of) p Fn 270 1690 a(L) p Fl 336 1705 a(1) p Ff 410 1690 a(indep) p 635 1690 a(endent) p 951 1690 a(of) p Fn 1065 1690 a(j) p Ff 1146 1690 a(such) p 1368 1690 a(that) p Fn 514 1899 a(D) p Fo 598 1899 a(\() p Fn(H) p Fm 717 1914 a(N) p Fo 784 1899 a(\)) p 850 1899 a(=) p Fi 953 1803 a(n) p Fn 1009 1899 a(u) p Fk 1092 1899 a(2) p Fn 1186 1899 a(D) p Fo 1270 1899 a(\() p Fn(L) p Fg 1374 1858 a(\003) p Fm 1374 1924 a(N) p Fo 1441 1899 a(\);) p Fn 1523 1899 a(t) p Fg 1558 1914 a(\000) p Fm(z) p Fe 1646 1924 a(j) p Fo 1682 1899 a(\() p Fn(\037) p Fm 1781 1914 a(j) p Fn 1818 1899 a(u) p Fo(\)) p Fk 1939 1899 a(2) p Fn 2033 1899 a(D) p Fo 2117 1899 a(\() p Fn(H) p Fl 2236 1914 a(1) p Fo 2275 1899 a(\)) p 2348 1899 a(for) p 2497 1899 a(ev) m(ery) p Fn 2756 1899 a(j) p Fo 2829 1899 a(=) p 2933 1899 a(1) p Fn(;) p 3026 1899 a(:) p 3070 1899 a(:) p 3114 1899 a(:) p 3156 1899 a(;) p 3200 1899 a(N) p Fi 3288 1803 a(o) p Fn 3361 1899 a(:) p Fo 3653 1899 a(\(1.14\)) p Ff 0 2108 a(Her) p 153 2108 a(e,) p Fn 263 2108 a(L) p Fg 329 2072 a(\003) p Fm 329 2133 a(N) p Ff 431 2108 a(is) p 536 2108 a(the) p 698 2108 a(adjoint) p 1029 2108 a(op) p 1124 2108 a(er) p 1205 2108 a(ator) p 1413 2108 a(of) p Fn 1527 2108 a(L) p Fm 1593 2123 a(N) p Ff 1661 2108 a(.) p Fo 0 2324 a(Remark) p 366 2324 a(1.) p 484 2324 a(The) p 684 2324 a(righ) m(t) p 918 2324 a(hand) p 1160 2324 a(side) p 1355 2324 a(of) p 1464 2324 a(\(1.14\)) p 1745 2324 a(is) p 1841 2324 a(indep) s(enden) m (t) p 2393 2324 a(of) p 2502 2324 a(the) p 2669 2324 a(c) m(hoice) p 2957 2324 a(of) p 3067 2324 a(the) p 3234 2324 a(function) p Fn 3614 2324 a(\037) p Fo 3706 2324 a(\(this) 0 2445 y(fact) p 192 2445 a(follo) m(ws) p 513 2445 a(from) p 743 2445 a(\(ii\)) p 906 2445 a(of) p 1017 2445 a(Prop) s(osition) p 1542 2445 a(4.2) p 1699 2445 a(b) s(elo) m(w\).) 0 2565 y(Remark) p 373 2565 a(2.) p 511 2565 a(There) p 799 2565 a(exists) p 1074 2565 a(a) p 1161 2565 a(self-adjoin) m(t) p 1669 2565 a(extension) p 2107 2565 a(of) p Fn 2224 2565 a(L) p Fm 2290 2580 a(N) p Fo 2396 2565 a(satisfying) p 2837 2565 a(\(1.14\)) p 3125 2565 a(for) p 3280 2565 a(an) m(y) p 3470 2565 a(giv) m(en) p 3731 2565 a(self-) 0 2685 y(adjoin) m(t) p 331 2685 a(extension) p Fn 762 2685 a(H) p Fl 843 2700 a(1) p Fo 915 2685 a(of) p Fn 1026 2685 a(L) p Fl 1092 2700 a(1) p Fo 1164 2685 a(\(see) p 1360 2685 a(Lemma) p 1708 2685 a(8.1\).) 146 2806 y(Our) p 347 2806 a(answ) m(er) p 670 2806 a(to) p 789 2806 a(the) p 957 2806 a(conjecture) p 1426 2806 a(\(I) s(I\)) p 1607 2806 a(is) p 1705 2806 a(the) p 1873 2806 a(follo) m(wing.) p Fq 146 3000 a(Theorem) p 621 3000 a(1.3) p Ff 813 3000 a(L) p 869 3000 a(et) p Fo 994 3000 a(1) p Fk 1094 3000 a(\024) p Fn 1223 3000 a(N) p 1363 3000 a(<) p Fk 1490 3000 a(1) p Ff 1637 3000 a(or) p Fn 1775 3000 a(N) p Fo 1915 3000 a(=) p Fk 2042 3000 a(1) p Ff(.) p 2254 3000 a(L) p 2310 3000 a(et) p Fn 2435 3000 a(H) p Fm 2516 3015 a(N) p Ff 2631 3000 a(b) p 2671 3000 a(e) p 2763 3000 a(a) p 2860 3000 a(self-adjoint) p 3379 3000 a(extension) p 3822 3000 a(of) p Fn 0 3120 a(L) p Fm 66 3135 a(N) p Ff 174 3120 a(which) p 453 3120 a(has) p 633 3120 a(the) p 800 3120 a(same) p 1055 3120 a(b) p 1095 3120 a(oundary) p 1480 3120 a(c) p 1520 3120 a(ondition) p 1911 3120 a(at) p 2033 3120 a(every) p Fn 2296 3120 a(z) p Fm 2341 3135 a(j) p Ff 2378 3120 a(.) p 2469 3120 a(L) p 2525 3120 a(et) p Fn 2643 3120 a(I) p Fo 2731 3120 a(=) p 2844 3120 a([) p Fn(c;) p 2957 3120 a(d) p Fo(]) p Ff 3075 3120 a(b) p 3115 3120 a(e) p 3200 3120 a(a) p 3290 3120 a(close) p 3490 3120 a(d) p 3579 3120 a(interval) 0 3241 y(satisfying) p 439 3241 a(that) p Fn 644 3241 a(I) p Fk 721 3241 a(\\) p 814 3241 a(f) p Fo(\(2) p Fn(n) p Fk 1035 3241 a(\000) p Fo 1139 3241 a(1\)) p Fn(B) p Fo 1305 3241 a(;) p Fn 1349 3241 a(n) p Fo 1445 3241 a(=) p 1559 3241 a(1) p Fn(;) p Fo 1652 3241 a(2) p Fn(;) p 1745 3241 a(:) p 1789 3241 a(:) p 1833 3241 a(:) p Fk 1859 3241 a(g) p Fo 1947 3241 a(=) p Fk 2061 3241 a(;) p Ff(,) p 2183 3241 a(that) p Fn 2388 3241 a(c;) p 2474 3241 a(d) p 2574 3241 a(=) p Fk 2562 3241 a(2) p Fn 2667 3241 a(\033) p Fo 2726 3241 a(\() p Fn(H) p Fl 2845 3256 a(1) p Fo 2884 3241 a(\)) p Ff 2963 3241 a(and) p 3158 3241 a(that) p Fn 3363 3241 a(\033) p Fo 3422 3241 a(\() p Fn(H) p Fl 3541 3256 a(1) p Fo 3580 3241 a(\)) p Fk 3644 3241 a(\\) p Fn 3737 3241 a(I) p Fo 3826 3241 a(=) p Fk 0 3361 a(f) p Fn(\025) p Fl 107 3376 a(1) p Fn 146 3361 a(;) p 190 3361 a(\025) p Fl 247 3376 a(2) p Fn 286 3361 a(;) p 330 3361 a(:) p 374 3361 a(:) p 418 3361 a(:) p 461 3361 a(;) p 505 3361 a(\025) p Fm 562 3376 a(k) p Fk 604 3361 a(g) p 682 3361 a(6) p Fo(=) p Fk 785 3361 a(;) p Ff(.) 146 3482 y(Then,) p 430 3482 a(ther) p 593 3482 a(e) p 671 3482 a(exist) p 897 3482 a(c) p 937 3482 a(onstants) p Fn 1325 3482 a(u) p 1408 3482 a(>) p Fo 1512 3482 a(0) p Ff 1594 3482 a(and) p Fn 1783 3482 a(R) p Fl 1857 3497 a(0) p Fn 1924 3482 a(>) p Fo 2028 3482 a(0) p Ff 2110 3482 a(dep) p 2250 3482 a(endent) p 2565 3482 a(on) p Fn 2704 3482 a(B) p Ff 2783 3482 a(,) p Fn 2847 3482 a(\013) p Ff 2910 3482 a(,) p Fn 2974 3482 a(I) p Ff 3025 3482 a(,) p Fn 3089 3482 a(H) p Fl 3170 3497 a(1) p Ff 3243 3482 a(\(indep) p 3508 3482 a(endent) p 3822 3482 a(of) p Fn 0 3602 a(N) p 83 3602 a(;) p 127 3602 a(R) p Ff 202 3602 a(\)) p 277 3602 a(satisfying) p 710 3602 a(the) p 872 3602 a(fol) p 982 3602 a(lowing:) 146 3722 y(\(i\)) p 291 3722 a(If) p Fn 393 3722 a(R) p Fk 496 3722 a(\025) p Fn 601 3722 a(R) p Fl 675 3737 a(0) p Ff 715 3722 a(,) p 779 3722 a(we) p 924 3722 a(have) p Fn 1296 4007 a(\033) p Fo 1355 4007 a(\() p Fn(H) p Fm 1474 4022 a(N) p Fo 1541 4007 a(\)) p Fk 1601 4007 a(\\) p Fn 1689 4007 a(I) p Fk 1768 4007 a(\032) p Fm 1910 3899 a(k) p Fi 1883 3924 a([) p Fm 1873 4108 a(l) p Fl 1895 4108 a(=1) p Fo 1985 4007 a([) p Fn(\025) p Fm 2069 4022 a(l) p Fk 2117 4007 a(\000) p Fn 2217 4007 a(\016) n(;) p 2302 4007 a(\025) p Fm 2359 4022 a(l) p Fo 2407 4007 a(+) p Fn 2505 4007 a(\016) p Fo 2552 4007 a(]) p Fn(;) p Ff 0 4305 a(wher) p 196 4305 a(e) p Fn 275 4305 a(\016) p Fo 350 4305 a(=) p Fn 453 4305 a(e) p Fg 498 4269 a(\000) p Fm(uR) p Fd 647 4246 a(2) p Ff 687 4305 a(.) 146 4426 y(\(ii\)) p 321 4426 a(If) p Fn 423 4426 a(R) p Fk 526 4426 a(\025) p Fn 631 4426 a(R) p Fl 705 4441 a(0) p Ff 745 4426 a(,) p 809 4426 a(we) p 954 4426 a(have) p Fo 1136 4635 a(dim) p 1315 4635 a(Ran) p Fn 1506 4635 a(P) p Fm 1569 4650 a(I) p Fo 1609 4635 a(\() p Fn(H) p Fm 1728 4650 a(N) p Fo 1795 4635 a(\)) p 1861 4635 a(=) p Fn 1964 4635 a(N) p Fo 2070 4635 a(dim) p 2249 4635 a(Ran) p Fn 2440 4635 a(P) p Fm 2503 4650 a(I) p Fo 2543 4635 a(\() p Fn(H) p Fl 2662 4650 a(1) p Fo 2701 4635 a(\)) p Fn(:) p Fo 0 4844 a(Note) p 236 4844 a(that) p Fn 447 4844 a(\033) p Fo 506 4844 a(\() p Fn(H) p Fl 625 4859 a(1) p Fo 664 4844 a(\)) p Fk 724 4844 a(\\) p Fn 813 4844 a(I) p Fo 896 4844 a(is) p 994 4844 a(a) p 1076 4844 a(\014nite) p 1325 4844 a(set) p 1477 4844 a(b) m(y) p 1613 4844 a(\(iii\)) p 1802 4844 a(of) p 1914 4844 a(Theorem) p 2325 4844 a(1.2.) 146 4965 y(Com) m(bining) p 642 4965 a(\(ii\)) p 805 4965 a(of) p 916 4965 a(Theorem) p 1328 4965 a(1.3) p 1485 4965 a(with) p 1707 4965 a(Theorem) p 2119 4965 a(1.1,) p 2303 4965 a(w) m(e) p 2447 4965 a(ha) m(v) m(e) p 2672 4965 a(the) p 2840 4965 a(follo) m(wing) p 3252 4965 a(corollary) p 3617 4965 a(.) 1927 5214 y(7) p eop %%Page: 8 8 8 7 bop Fq 146 631 a(Corollary) p 644 631 a(1.4) p Ff 836 631 a(L) p 892 631 a(et) p Fo 1013 631 a(1) p Fk 1105 631 a(\024) p Fn 1225 631 a(N) p 1357 631 a(<) p Fk 1476 631 a(1) p Ff 1619 631 a(and) p 1816 631 a(let) p Fn 1962 631 a(\013) p Fl 2024 646 a(1) p Fo 2106 631 a(=) p Fn 2225 631 a(\013) p Fl 2287 646 a(2) p Fo 2370 631 a(=) p Fk 2489 631 a(\001) p 2534 631 a(\001) p 2579 631 a(\001) p Fo 2648 631 a(=) p Fn 2767 631 a(\013) p Fm 2829 646 a(N) p Fo 2939 631 a(=) p Fn 3058 631 a(\013) p Ff 3121 631 a(.) p 3221 631 a(Then,) p 3515 631 a(for) p 3679 631 a(every) p Fn 0 751 a(n) p Fl 58 766 a(0) p Fo 128 751 a(=) p 235 751 a(1) p Fn(;) p Fo 328 751 a(2) p Fn(;) p 421 751 a(:) p 465 751 a(:) p 509 751 a(:) p Ff 534 751 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a(\() p Fn(H) p Fm 2212 1462 a(n) 2204 1527 y(N) p Fo 2271 1503 a(\)) p 2392 1503 a(=) p 2551 1503 a(\() p Fn(n) p Fo 2669 1503 a(+) p 2767 1503 a(1\)) p Fn(N) p Ff 0 1723 a(for) p Fn 156 1723 a(n) p Fo 242 1723 a(=) p 345 1723 a(1) p Fn(;) p Fo 438 1723 a(2) p Fn(;) p 531 1723 a(:) p 575 1723 a(:) p 619 1723 a(:) p 661 1723 a(;) p 705 1723 a(n) p Fl 763 1738 a(0) p Ff 803 1723 a(.) p Fo 146 1926 a(W) p 238 1926 a(e) p 314 1926 a(shall) p 542 1926 a(shortly) p 871 1926 a(exhibit) p 1199 1926 a(the) p 1367 1926 a(outline) p 1692 1926 a(of) p 1803 1926 a(the) p 1971 1926 a(pro) s(of) p 2226 1926 a(of) p 2337 1926 a(our) p 2510 1926 a(theorems.) 146 2046 y(In) p 273 2046 a(the) p 445 2046 a(pro) s(of) p 704 2046 a(of) p 820 2046 a(Theorem) p 1236 2046 a(1.2,) p 1426 2046 a(the) p 1598 2046 a(canonical) p 2031 2046 a(comm) m(utation) p 2628 2046 a(relation) p 2991 2046 a(\(CCR\)) p 3316 2046 a(of) p 3432 2046 a(the) p 3604 2046 a(annihi-) 0 2167 y(lation) p 280 2167 a(op) s(erator) p Fn 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a(case,) 0 2528 y(CCR) p 244 2528 a(holds) p 498 2528 a(with) p 719 2528 a(a) p 799 2528 a(p) s(erturbation) p 1369 2528 a(b) m(y) p Fn 1504 2528 a(\016) p Fo 1551 2528 a(-lik) m(e) p 1761 2528 a(magnetic) p 2177 2528 a(\014elds.) p 2464 2528 a(This) p 2685 2528 a(p) s(erturbation) p 3256 2528 a(mak) m(es) p 3548 2528 a(t) m(w) m(o) p 3731 2528 a(self-) 0 2666 y(adjoin) m(t) p 323 2666 a(op) s(erators) p 747 2666 a(\() p Fn(A) p Fg 858 2618 a(y) p Fm 858 2690 a(N) p Fo 925 2666 a(\)) p Fg 963 2629 a(\003) p 1002 2562 141 4 v Fn 1002 2666 a(A) p Fg 1075 2618 a(y) p Fm 1075 2690 a(N) p Fk 1149 2666 a(\000) p Fn 1232 2666 a(B) p Fo 1337 2666 a(and) p 1519 2562 V Fn 1519 2666 a(A) p Fg 1592 2618 a(y) p Fm 1592 2690 a(N) p Fo 1660 2666 a(\() p Fn(A) p Fg 1771 2618 a(y) p Fm 1771 2690 a(N) p Fo 1838 2666 a(\)) p Fg 1876 2629 a(\003) p Fo 1922 2666 a(+) p Fn 2004 2666 a(B) p Fo 2108 2666 a(di\013eren) m(t) p 2485 2666 a(\(the) p 2683 2666 a(o) m(v) m(erline) p 3036 2666 a(denotes) p 3381 2666 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a(t) m(yp) s(e,) p 3193 3027 a(to) p 3312 3027 a(determine) p 3767 3027 a(the) 0 3147 y(essen) m(tial) p 392 3147 a(sp) s(ectrum) p 821 3147 a(of) p 935 3147 a(the) p 1106 3147 a(Sc) m(hr\177) p 1292 3147 a(odinger) p 1644 3147 a(op) s(erators) p 2078 3147 a(on) p Fj 2217 3147 a(R) p Fl 2304 3105 a(2) p Fo 2379 3147 a(with) p 2605 3147 a(the) p 2776 3147 a(magnetic) p 3196 3147 a(\014elds) p 3449 3147 a(con) m(v) m(erging) 0 3268 y(to) p 119 3268 a(a) p 201 3268 a(non-zero) p 596 3268 a(constan) m(t) p 990 3268 a(at) p 1109 3268 a(in\014nit) m(y) p 1405 3268 a(.) 146 3388 y(Theorem) p 559 3388 a(1.3) p 716 3388 a(is) p 814 3388 a(an) p 950 3388 a(analogy) p 1311 3388 a(of) p 1422 3388 a(the) p 1591 3388 a(result) p 1862 3388 a(of) p 1974 3388 a(Cornean) p 2364 3388 a(and) p 2554 3388 a(Nenciu) p 2883 3388 a(\([Co-Ne,) p 3276 3388 a(Theorem) p 3688 3388 a(I) s(I) s(I.1,) 0 3508 y(Corollary) p 431 3508 a(I) s(I) s(I.1]\).) p 753 3508 a(They) p 1005 3508 a(treat) p 1244 3508 a(the) p 1412 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p 2793 4790 a(v) m(ector) p 3093 4790 a(p) s(oten) m(tials) p 3552 4790 a(\(see) p 3756 4790 a([Ik-) 0 4910 y(Ka,) p 191 4910 a(Theorem) p 608 4910 a(1]) p 722 4910 a(or) p 846 4910 a([Le-Si,) p 1158 4910 a(Theorem) p 1575 4910 a(2]\).) p 1775 4910 a(Th) m(us) p 2027 4910 a(w) m(e) p 2176 4910 a(do) p 2317 4910 a(not) p 2496 4910 a(use) p 2669 4910 a(the) p 2843 4910 a(appro) m(ximating) p 3498 4910 a(argumen) m(t) 1927 5214 y(8) p eop %%Page: 9 9 9 8 bop Fn 0 631 a(N) p Fk 116 631 a(!) p 243 631 a(1) p Fo(,) p 401 631 a(whic) m(h) p 677 631 a(is) p 773 631 a(used) p 993 631 a(in) p 1104 631 a(their) p 1335 631 a(pap) s(er.) p 1646 631 a(W) p 1738 631 a(e) p 1811 631 a(pro) m(v) m(e) p 2071 631 a(the) p 2237 631 a(statemen) m(t) p 2687 631 a(of) p 2796 631 a(Theorem) p 3205 631 a(1.3) p 3360 631 a(directly) p 3712 631 a(ev) m(en) 0 751 y(when) p Fn 255 751 a(N) p Fo 371 751 a(=) p Fk 474 751 a(1) p Fo(,) p 634 751 a(using) p 889 751 a(the) p 1057 751 a(argumen) m(t) p 1493 751 a(of) p 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a(simpli\014es) p 2693 2116 a(our) p 2867 2116 a(calculus) p 3236 2116 a(in) p 3349 2116 a(man) m(y) p 3615 2116 a(cases.) p Fq 1512 2391 a(Ac) m(kno) m(wledgmen) m(ts) p Fo 146 2619 a(The) p 363 2619 a(author) p 693 2619 a(w) m(ould) p 994 2619 a(lik) m(e) p 1188 2619 a(to) p 1324 2619 a(thank) p 1619 2619 a(Akira) p 1906 2619 a(Iw) m(atsuk) p 2233 2619 a(a) p 2331 2619 a(and) p 2537 2619 a(all) p 2688 2619 a(other) p 2959 2619 a(mem) m(b) s(ers) p 3387 2619 a(of) p 3515 2619 a(Op) s(erator) 0 2740 y(Seminar) p 395 2740 a(in) p 524 2740 a(RIMS,) p 849 2740 a(for) p 1014 2740 a(man) m(y) p 1295 2740 a(helpful) p 1632 2740 a(commen) m(ts.) p 2175 2740 a(In) p 2312 2740 a(particular,) p 2808 2740 a(the) p 2992 2740 a(author) p 3321 2740 a(w) m(ould) p 3621 2740 a(lik) m(e) p 3815 2740 a(to) 0 2860 y(thank) p 283 2860 a(Shin-ic) m(hi) p 691 2860 a(Shirai) p 977 2860 a(for) p 1130 2860 a(giving) p 1424 2860 a(the) p 1596 2860 a(author) p 1914 2860 a(an) p 2054 2860 a(opp) s(ortunit) m (y) 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2479 4097 a(=) p 2584 4097 a(1) p Fn(;) p Fo 2677 4097 a(2) p Fn(;) p 2770 4097 a(:) p 2814 4097 a(:) p 2858 4097 a(:) p Ff 2919 4097 a(or) p Fn 3046 4097 a(N) p Fo 3163 4097 a(=) p Fk 3268 4097 a(1) p Ff 3404 4097 a(and) p Fk 3594 4097 a(f) p Fn(z) p Fm 3689 4112 a(j) p Fk 3725 4097 a(g) p Fm 3775 4061 a(N) 3775 4122 y(j) p Fl 3808 4122 a(=1) p Ff 0 4217 a(b) p 40 4217 a(e) p 122 4217 a(\(at) p 282 4217 a(most\)) p 561 4217 a(c) p 601 4217 a(ountable) p 992 4217 a(p) p 1037 4217 a(oints) p 1281 4217 a(in) p Fn 1403 4217 a(U) p Ff 1517 4217 a(with) p 1731 4217 a(no) p 1873 4217 a(ac) p 1963 4217 a(cumulation) p 2471 4217 a(p) p 2516 4217 a(oints) p 2760 4217 a(in) p Fn 2882 4217 a(U) p Ff 2958 4217 a(.) p 3041 4217 a(Put) p Fn 3230 4217 a(S) p Fm 3290 4232 a(N) p Fo 3389 4217 a(=) p Fk 3498 4217 a([) p Fm 3564 4181 a(N) 3564 4242 y(j) p Fl 3597 4242 a(=1) p Fk 3691 4217 a(f) p Fn(z) p Fm 3786 4232 a(j) p Fk 3822 4217 a(g) p Ff(.) 0 4338 y(L) p 56 4338 a(et) p Fk 172 4338 a(f) p 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4468 a(the) p 2102 4468 a(e) p 2142 4468 a(quality) p Fo 1143 4763 a(rot) p Fj 1284 4763 a(a) p Fo(\() p Fn(z) p Fo 1434 4763 a(\)) p 1500 4763 a(=) p Fm 1634 4655 a(N) p Fi 1605 4680 a(X) p Fm 1604 4862 a(j) p Fl 1637 4862 a(=1) p Fo 1743 4763 a(2) p Fn(\031) t(\013) p Fm 1913 4778 a(j) p Fn 1949 4763 a(\016) p Fo 1996 4763 a(\() p Fn(z) p Fk 2106 4763 a(\000) p Fn 2206 4763 a(z) p Fm 2251 4778 a(j) p Fo 2288 4763 a(\)) p 2360 4763 a(in) p Fk 2477 4763 a(D) p Fg 2557 4722 a(0) p Fo 2580 4763 a(\() p Fn(U) p Fo 2694 4763 a(\)) p Fn(:) p Fo 3701 4763 a(\(2.1\)) 1927 5214 y(9) p eop %%Page: 10 10 10 9 bop Fo 0 631 a(Th) m(us) p 244 631 a(w) m(e) p 385 631 a(can) p 561 631 a(construct) p 986 631 a(a) p 1064 631 a(v) m(ector) p 1354 631 a(p) s(oten) m(tial) p Fj 1763 631 a(a) p Fm 1826 646 a(N) p Fo 1923 631 a(satisfying) p 2354 631 a(\(1.2\)) p 2584 631 a(b) m(y) p 2717 631 a(putting) p Fj 3061 631 a(a) p Fm 3124 646 a(N) p Fo 3219 631 a(=) p Fj 3322 631 a(a) p Fl 3385 646 a(0) p Fo 3441 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2874 4350 a(1) p Fn(;) p Fo 2967 4350 a(2) p Fn(;) p 3060 4350 a(:) p 3104 4350 a(:) p 3148 4350 a(:) p Fo 0 4604 a(holds,) p 282 4604 a(one) p 461 4604 a(can) p 640 4604 a(tak) m(e) p Fn 580 4877 a(\036) p Fo(\() p Fn(z) p Fo 725 4877 a(\)) p 791 4877 a(=) p Fn 904 4810 a(\013) p Fl 966 4825 a(1) p 904 4854 102 4 v Fn 930 4946 a(z) p Fo 1038 4877 a(+) p Fg 1162 4770 a(1) p Fi 1137 4794 a(X) p Fm 1136 4977 a(j) p Fl 1169 4977 a(=2) p Fn 1275 4877 a(\013) p Fm 1337 4892 a(j) p Fi 1390 4731 a( ) p Fo 1568 4810 a(1) p 1466 4854 254 4 v Fn 1466 4946 a(z) p Fk 1538 4946 a(\000) p Fn 1638 4946 a(z) p Fm 1683 4961 a(j) p Fo 1751 4877 a(+) 1876 4810 y(1) p 1859 4854 82 4 v Fn 1859 4946 a(z) p Fm 1904 4961 a(j) p Fo 1973 4877 a(+) p Fn 2101 4810 a(z) p 2081 4854 90 4 v 2081 4946 a(z) p Fl 2130 4911 a(2) p Fm 2126 4969 a(j) p Fi 2180 4731 a(!) p Fo 2274 4877 a(=) p Fn 2387 4810 a(\013) p Fl 2449 4825 a(1) p 2387 4854 102 4 v Fn 2413 4946 a(z) p Fo 2521 4877 a(+) p Fg 2645 4770 a(1) p Fi 2620 4794 a(X) p Fm 2619 4977 a(j) p Fl 2652 4977 a(=2) p Fn 2758 4877 a(\013) p Fm 2820 4892 a(j) p Fn 3031 4810 a(z) p Fl 3080 4774 a(2) p 2867 4854 419 4 v Fo 2867 4946 a(\() p Fn(z) p Fk 2976 4946 a(\000) p Fn 3076 4946 a(z) p Fm 3121 4961 a(j) p Fo 3158 4946 a(\)) p Fn(z) p Fl 3245 4911 a(2) p Fm 3241 4969 a(j) p Fn 3295 4877 a(:) p Fo 1902 5214 a(10) p eop %%Page: 11 11 11 10 bop Fo 0 631 a(The) p 199 631 a(sum) p 405 631 a(con) m(v) m(erges) p 841 631 a(absolutely) p 1265 631 a(,) p 1323 631 a(uniformly) p 1766 631 a(in) p 1879 631 a(a) p 1959 631 a(compact) p 2348 631 a(set) p Fn 2500 631 a(K) p Fo 2621 631 a(in) p Fj 2734 631 a(R) p Fl 2821 589 a(2) p Fk 2881 631 a(n) p Fn 2951 631 a(S) p Fg 3011 646 a(1) p Fo 3085 631 a(,) p 3144 631 a(since) p 3382 631 a(the) p 3549 631 a(absolute) 0 751 y(v) p 46 751 a(alue) p 259 751 a(of) p 378 751 a(the) p 553 751 a(summand) p 1006 751 a(is) p Fn 1111 751 a(O) p Fo 1189 751 a(\() p Fk(j) p Fn(z) p Fm 1300 766 a(j) p Fk 1336 751 a(j) p Fg 1364 715 a(\000) p Fl(3) p Fo 1458 751 a(\)) p 1536 751 a(uniformly) p 1987 751 a(with) p 2217 751 a(resp) s(ect) p 2558 751 a(to) p Fn 2685 751 a(z) p Fk 2775 751 a(2) p Fn 2882 751 a(K) p Fo 2972 751 a(.) p 3064 751 a(In) p 3194 751 a(particular,) p 3680 751 a(when) p Fn 0 872 a(S) p Fg 60 887 a(1) p Fo 172 872 a(is) p 275 872 a(a) p 362 872 a(lattice) p 665 872 a(and) p 859 872 a(ev) m(ery) p Fn 1122 872 a(\013) p Fm 1184 887 a(j) p Fo 1258 872 a(tak) m(es) p 1513 872 a(the) p 1685 872 a(same) p 1935 872 a(v) p 1981 872 a(alue) p Fn 2192 872 a(\013) p Fo 2255 872 a(,) p 2320 872 a(the) p 2493 872 a(function) p Fn 2880 872 a(\036) p Fo 2976 872 a(equals) p 3276 872 a(to) p Fn 3401 872 a(\013) p Fo 3501 872 a(times) p 3767 872 a(the) 0 992 y(W) p 92 992 a(eierstrass) p Fn 516 992 a(\020) p Fo 599 992 a(function.) 146 1112 y(In) p 269 1112 a(the) p 438 1112 a(previous) p 827 1112 a(lemma,) p 1170 1112 a(w) m(e) p 1315 1112 a(tak) m(e) p 1527 1112 a(a) p 1610 1112 a(sp) s(ecial) p 1928 1112 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Fn(;) p 3417 1425 a(:) p 3461 1425 a(:) p 3505 1425 a(:) p Ff 3570 1425 a(or) p Fn 3701 1425 a(N) p Fo 3826 1425 a(=) p Fk 0 1545 a(1) p Ff(.) p 189 1545 a(L) p 245 1545 a(et) p Fk 362 1545 a(f) p Fn(z) p Fm 457 1560 a(j) p Fk 494 1545 a(g) p Fm 544 1509 a(N) 544 1570 y(j) p Fl 577 1570 a(=1) p Ff 710 1545 a(ar) p 796 1545 a(e) p 880 1545 a(at) p 1002 1545 a(most) p 1244 1545 a(c) p 1284 1545 a(ountable) p 1677 1545 a(p) p 1722 1545 a(oints) p 1968 1545 a(in) p Fn 2092 1545 a(U) p Ff 2209 1545 a(with) p 2425 1545 a(no) p 2570 1545 a(ac) p 2660 1545 a(cumulation) p 3170 1545 a(p) p 3215 1545 a(oints) p 3461 1545 a(in) p Fn 3585 1545 a(U) p Ff 3661 1545 a(.) p 3751 1545 a(Put) p Fn 0 1676 a(S) p Fm 60 1691 a(N) p Fo 155 1676 a(=) p Fk 258 1676 a([) p Fm 324 1640 a(N) 324 1701 y(j) p Fl 357 1701 a(=1) p Fk 452 1676 a(f) p Fn(z) p Fm 547 1691 a(j) p Fk 583 1676 a(g) p Ff(.) p 708 1676 a(L) p 764 1676 a(et) p Fj 876 1676 a(a) p Fn(;) p Fj 983 1676 a(b) p Fk 1062 1676 a(2) p Fn 1156 1676 a(C) p Fg 1233 1640 a(1) p Fo 1308 1676 a(\() p Fn(U) p Fk 1445 1676 a(n) p Fn 1517 1676 a(S) p Fm 1577 1691 a(N) p Fo 1644 1676 a(;) p Fj 1688 1676 a(R) p Fl 1775 1634 a(2) p Fo 1814 1676 a(\)) p Fk 1874 1676 a(\\) p Fn 1963 1676 a(L) p Fl 2029 1640 a(1) p Fm 2029 1701 a(l) q(oc) p Fo 2120 1676 a(\() p Fn(U) p Fo 2234 1676 a(;) p Fj 2278 1676 a(R) p Fl 2366 1634 a(2) p Fo 2405 1676 a(\)) p Ff 2478 1676 a(and) p 2667 1676 a(assume) p 3008 1676 a(that) p Fo 1470 1883 a(rot) p Fj 1611 1883 a(a) p Fo 1702 1883 a(=) p 1805 1883 a(rot) p Fj 1947 1883 a(b) p Fo 2033 1883 a(in) p Fk 2150 1883 a(D) p Fg 2230 1842 a(0) p Fo 2252 1883 a(\() p Fn(U) p Fo 2366 1883 a(\)) p Fn(:) p Ff 0 2090 a(Then,) p 276 2090 a(ther) p 439 2090 a(e) p 509 2090 a(exists) p 765 2090 a(a) p 840 2090 a(multiplic) p 1204 2090 a(ation) p 1445 2090 a(op) p 1540 2090 a(er) p 1621 2090 a(ator) p 1819 2090 a(by) p 1936 2090 a(a) p 2011 2090 a(c) p 2051 2090 a(omplex-value) p 2593 2090 a(d) p 2666 2090 a(smo) p 2831 2090 a(oth) p 2987 2090 a(function) p Fo 3361 2090 a(\010) p Fk 3459 2090 a(2) p Fn 3553 2090 a(C) p Fg 3630 2053 a(1) p Fo 3705 2090 a(\() p Fj(R) p Fl 3830 2048 a(2) p Fk 3869 2090 a(n) p Fn 0 2210 a(S) p Fm 60 2225 a(N) p Fo 127 2210 a(\)) p Ff 200 2210 a(with) p Fk 412 2210 a(j) p Fo(\010\() p Fn(z) p Fo 597 2210 a(\)) p Fk(j) p Fo 691 2210 a(=) p 794 2210 a(1) p Ff 878 2210 a(for) p 1034 2210 a(every) p Fn 1291 2210 a(z) p Fk 1369 2210 a(2) p Fj 1463 2210 a(R) p Fl 1550 2168 a(2) p Fk 1612 2210 a(n) p Fn 1684 2210 a(S) p Fm 1744 2225 a(N) p Ff 1811 2210 a(,) p 1876 2210 a(such) p 2098 2210 a(that) p 2297 2210 a(the) p 2459 2210 a(e) p 2499 2210 a(quality) p Fo 1436 2417 a(\010) p Fg 1506 2376 a(\000) p Fl(1) p Fo 1601 2417 a(\() p Fk(r) p Fo 1744 2417 a(+) p Fn 1842 2417 a(i) p Fj(b) p Fo(\)\010) p 2063 2417 a(=) p Fk 2167 2417 a(r) p Fo 2272 2417 a(+) p Fn 2370 2417 a(i) p Fj(a) p Fo 3701 2417 a(\(2.2\)) p Ff 0 2624 a(holds) p 249 2624 a(as) p 374 2624 a(an) p 513 2624 a(op) p 608 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a(p) s(oin) m(t) p Fn 657 3537 a(z) p Fl 702 3552 a(0) p Fk 770 3537 a(2) p Fn 864 3537 a(U) p Fk 962 3537 a(n) p Fn 1034 3537 a(S) p Fm 1094 3552 a(N) p Fo 1162 3537 a(.) p 1232 3537 a(F) p 1288 3537 a(or) p Fn 1407 3537 a(z) p Fk 1484 3537 a(2) p Fn 1578 3537 a(U) p Fk 1677 3537 a(n) p Fn 1749 3537 a(S) p Fm 1809 3552 a(N) p Fo 1876 3537 a(,) p 1936 3537 a(de\014ne) p Fn 1382 3774 a(\036) p Fo(\() p Fn(z) p Fo 1527 3774 a(\)) p 1593 3774 a(=) p Fi 1696 3657 a(Z) p Fm 1742 3846 a(\015) p Fe 1778 3854 a(z) p Fd 1808 3869 a(0) p Fe 1843 3854 a(;z) p Fo 1902 3774 a(\() p Fn(a) p Fm 1991 3789 a(x) p Fn 2035 3774 a(dx) p Fo 2163 3774 a(+) p Fn 2261 3774 a(a) p Fm 2312 3789 a(y) p Fn 2354 3774 a(dy) p Fo 2457 3774 a(\)) p Fn(;) p Fo 0 4033 a(where) p Fn 282 4033 a(\015) p Fm 333 4048 a(z) p Fd 366 4057 a(0) p Fm 400 4048 a(;z) p Fo 492 4033 a(is) p 590 4033 a(a) p 671 4033 a(piecewise) p 1094 4033 a(linear) p 1365 4033 a(path) p 1593 4033 a(from) p Fn 1823 4033 a(z) p Fl 1868 4048 a(0) p Fo 1941 4033 a(to) p Fn 2060 4033 a(z) p Fo 2142 4033 a(not) p 2315 4033 a(crossing) p 2685 4033 a(the) p 2853 4033 a(p) s(oin) m(ts) p 3146 4033 a(in) p Fn 3260 4033 a(S) p Fm 3320 4048 a(N) p Fo 3387 4033 a(.) 146 4153 y(W) p 238 4153 a(e) p 310 4153 a(shall) p 535 4153 a(see) p 689 4153 a(that) p Fn 896 4153 a(\036) p Fo(\() p Fn(z) p Fo 1041 4153 a(\)) p 1108 4153 a(is) p 1203 4153 a(indep) s(enden) m(t) p 1752 4153 a(of) p 1859 4153 a(the) p 2023 4153 a(c) m(hoice) p 2309 4153 a(of) p 2417 4153 a(paths) p Fn 2679 4153 a(\015) p Fm 2730 4168 a(z) p Fd 2763 4177 a(0) p Fm 2797 4168 a(;z) p Fo 2856 4153 a(.) p 2925 4153 a(T) p 2987 4153 a(ak) m(e) p 3157 4153 a(another) p 3511 4153 a(piecewise) 0 4274 y(linear) p 269 4274 a(path) p 499 4274 a(~) p Fn 496 4274 a(\015) p Fm 547 4289 a(z) p Fd 580 4298 a(0) p Fm 614 4289 a(;z) p Fo 704 4274 a(from) p Fn 933 4274 a(z) p Fl 978 4289 a(0) p Fo 1049 4274 a(to) p Fn 1166 4274 a(z) p Fo 1215 4274 a(.) p 1286 4274 a(W) p 1378 4274 a(e) p 1452 4274 a(ma) m(y) p 1662 4274 a(assume) p 1998 4274 a(that) p 2208 4274 a(t) m(w) m(o) p 2391 4274 a(paths) p 2655 4274 a(cross) p 2893 4274 a(only) p 3106 4274 a(at) p Fn 3223 4274 a(z) p Fl 3268 4289 a(0) p Fo 3339 4274 a(and) p Fn 3527 4274 a(z) p Fo 3608 4274 a(\(if) p 3734 4274 a(not,) 0 4394 y(divide) p 296 4394 a(paths) p 568 4394 a(b) m(y) p 710 4394 a(their) p 949 4394 a(crossing) p 1325 4394 a(p) s(oin) m(ts) p 1624 4394 a(and) p 1820 4394 a(consider) p 2206 4394 a(eac) m(h) p 2431 4394 a(divided) p 2782 4394 a(paths\).) p 3142 4394 a(Since) p Fn 3403 4394 a(U) p Fo 3518 4394 a(is) p 3622 4394 a(simply) 0 4514 y(connected,) p 489 4514 a(the) p 662 4514 a(union) p 938 4514 a(of) p 1055 4514 a(t) m(w) m(o) p 1244 4514 a(paths) p 1515 4514 a(is) p 1619 4514 a(the) p 1792 4514 a(b) s(oundary) p 2236 4514 a(of) p 2353 4514 a(an) p 2493 4514 a(op) s(en) p 2734 4514 a(set) p 2892 4514 a(\012) p 3000 4514 a(con) m(tained) p 3447 4514 a(in) p Fn 3566 4514 a(U) p Fo 3642 4514 a(.) p 3729 4514 a(If) p 3832 4514 a(\012) 0 4635 y(do) s(es) p 220 4635 a(not) p 393 4635 a(con) m(tain) p 738 4635 a(p) s(oin) m(ts) p 1031 4635 a(in) p Fn 1145 4635 a(S) p Fm 1205 4650 a(N) p Fo 1272 4635 a(,) p 1331 4635 a(then) p 1554 4635 a(rot) p Fj 1695 4635 a(a) p Fo 1790 4635 a(is) p 1889 4635 a(smo) s(oth) p 2233 4635 a(in) p 2347 4635 a(\012) p 2450 4635 a(and) p 2640 4635 a(w) m(e) p 2783 4635 a(ha) m(v) m(e) p Fi 929 4752 a( ) 995 4781 y(Z) p Fm 1041 4970 a(\015) p Fe 1077 4978 a(z) p Fd 1107 4993 a(0) p Fe 1141 4978 a(;z) p Fk 1217 4898 a(\000) p Fi 1311 4781 a(Z) p Fl 1360 4970 a(~) p Fm 1357 4970 a(\015) p Fe 1393 4978 a(z) p Fd 1423 4993 a(0) p Fe 1458 4978 a(;z) p Fi 1517 4752 a(!) p Fo 1599 4898 a(\() p Fn(a) p Fm 1688 4913 a(x) p Fn 1732 4898 a(dx) p Fo 1860 4898 a(+) p Fn 1958 4898 a(a) p Fm 2009 4913 a(y) p Fn 2051 4898 a(dy) p Fo 2154 4898 a(\)) p 2218 4898 a(=) p Fk 2322 4898 a(\006) p Fi 2416 4781 a(Z) p Fl 2462 4970 a(\012) p Fo 2534 4898 a(rot) p Fj 2675 4898 a(a) p Fn(dxdy) t(;) p Fo 3701 4898 a(\(2.4\)) 1902 5214 y(11) p eop %%Page: 12 12 12 11 bop Fo 0 631 a(b) m(y) p 137 631 a(the) p 306 631 a(Stok) m(es) p 611 631 a(theorem) p 992 631 a(\(the) p 1199 631 a(sign) p Fk 1402 631 a(\006) p Fo 1513 631 a(is) p 1612 631 a(determined) p 2123 631 a(b) m(y) p 2260 631 a(the) p 2429 631 a(direction) p 2837 631 a(of) p 2949 631 a(paths) p Fn 3216 631 a(\015) p Fm 3267 646 a(z) p Fd 3300 655 a(0) p Fm 3334 646 a(;z) p Fk 3417 631 a([) p Fo 3506 631 a(\() p Fk(\000) p Fo 3625 631 a(~) p Fn 3621 631 a(\015) p Fm 3672 646 a(z) p Fd 3705 655 a(0) p Fm 3740 646 a(;z) p Fo 3799 631 a(\)\).) 0 751 y(By) p 153 751 a(\(2.3\),) p 413 751 a(w) m(e) p 557 751 a(ha) m(v) m(e) p Fi 1075 785 a(Z) p Fm 1121 973 a(\015) p Fe 1157 981 a(z) p Fd 1187 996 a(0) p Fe 1221 981 a(;z) p Fo 1280 902 a(\() p Fn(a) p Fm 1369 917 a(x) p Fn 1413 902 a(dx) p Fo 1541 902 a(+) p Fn 1639 902 a(a) p Fm 1690 917 a(y) p Fn 1732 902 a(dy) p Fo 1835 902 a(\)) p 1900 902 a(=) p Fi 2003 785 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4504 a(\010\() p Fn(z) p Fo 1608 4504 a(\)) p Fg 1646 4463 a(\000) p Fl(1) p Fk 1741 4504 a(r) p Fo(\010\() p Fn(z) p Fo 1981 4504 a(\)) p 2048 4504 a(=) p Fk 2152 4504 a(r) p Fo 2257 4504 a(+) p Fn 2355 4504 a(i) p Fj(a) p Fo 3701 4504 a(\(2.6\)) p Ff 0 4724 a(as) p 125 4724 a(an) p 264 4724 a(op) p 359 4724 a(er) p 440 4724 a(ator) p 648 4724 a(fr) p 714 4724 a(om) p Fk 878 4724 a(D) p Fg 958 4688 a(0) p Fo 981 4724 a(\() p Fn(U) p Fk 1118 4724 a(n) p Fn 1190 4724 a(S) p Fm 1250 4739 a(N) p Fo 1317 4724 a(\)) p Ff 1390 4724 a(to) p Fo 1507 4724 a(\() p Fk(D) p Fg 1625 4688 a(0) p Fo 1648 4724 a(\() p Fn(U) p Fk 1784 4724 a(n) p Fn 1856 4724 a(S) p Fm 1916 4739 a(N) p Fo 1983 4724 a(\)\)) p Fl 2059 4688 a(2) p Ff 2099 4724 a(.) p Fo 1902 5214 a(12) p eop %%Page: 13 13 13 12 bop Ff 146 631 a(Pr) p 248 631 a(o) p 293 631 a(of.) p Fo 448 631 a(Assume) p 822 631 a(all) p Fn 970 631 a(\013) p Fm 1032 646 a(j) p Fo 1069 631 a('s) p 1179 631 a(are) p 1355 631 a(in) m(tegers.) p 1791 631 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p Fo 2259 872 a(\)) p Fk(j) p Fo 2353 872 a(=) p 2457 872 a(1,) p 2565 872 a(since) p Fi 1426 981 a(I) p Fm 1472 1170 a(C) p Fe 1522 1180 a(j) p Fn 1575 1098 a(\036) p Fo(\() p Fn(z) p Fo 1720 1098 a(\)) p 1787 1098 a(=) p 1890 1098 a(2) p Fn(\031) t(i\013) p Fm 2093 1113 a(j) p Fk 2157 1098 a(2) p Fo 2251 1098 a(2) p Fn(\031) t(i) p Fj(Z) p Fo 0 1343 a(for) p 149 1343 a(an) m(y) p 333 1343 a(small) p 589 1343 a(circular) p 941 1343 a(path) p Fn 1168 1343 a(C) p Fm 1238 1358 a(j) p Fo 1307 1343 a(going) p 1567 1343 a(around) p Fn 1898 1343 a(z) p Fm 1943 1358 a(j) p Fo 2012 1343 a(coun) m(terclo) s(c) m(kwise.) p 2793 1343 a(Then) p 3048 1343 a(w) m(e) p 3192 1343 a(ha) m(v) m(e) 1286 1540 y(\010\() p Fn(z) p Fo 1443 1540 a(\)) p Fg 1481 1498 a(\000) p Fl(1) p Fk 1577 1540 a(r) p Fo(\010\() p Fn(z) p Fo 1817 1540 a(\)) 1128 1685 y(=) p Fk 1286 1685 a(r) p Fo 1392 1685 a(+) p Fn 1490 1685 a(i) p Fo(\() p Fn(@) p Fm 1612 1700 a(x) p Fo 1673 1685 a(Im) p Fn 1806 1685 a( ) p Fo 1873 1685 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m(y-Riemann) p 1579 2172 a(relation.) p 1975 2104 64 4 v 1975 2197 4 93 v 2035 2197 V 1975 2200 64 4 v 146 2373 a(The) p 341 2373 a(existence) p 750 2373 a(of) p 855 2373 a(the) p 1018 2373 a(magnetic) p 1429 2373 a(translation) p 1917 2373 a(op) s(erator) p 2304 2373 a(de\014ned) p 2634 2373 a(in) p 2742 2373 a(De\014nition) p 3190 2373 a(1.1) p 3341 2373 a(is) p 3433 2373 a(guaran) m(teed) 0 2494 y(b) m(y) p 135 2494 a(the) p 303 2494 a(follo) m(wing) p 715 2494 a(more) p 959 2494 a(general) p 1295 2494 a(statemen) m(t.) p Fq 146 2677 a(Prop) s(osition) p 752 2677 a(2.4) p Ff 944 2677 a(L) p 1000 2677 a(et) p Fn 1126 2677 a(U) p Ff 1251 2677 a(and) p Fn 1454 2677 a(V) p Ff 1581 2677 a(ar) p 1667 2677 a(e) p 1760 2677 a(op) p 1855 2677 a(en) p 2003 2677 a(sets) p 2208 2677 a(in) p Fj 2341 2677 a(R) p Fl 2428 2635 a(2) p Ff 2516 2677 a(satisfying) p Fn 2963 2677 a(V) p Fo 3095 2677 a(=) p Fn 3223 2677 a(U) p Fo 3332 2677 a(+) p Fn 3440 2677 a(w) p Fo 3566 2677 a(=) p Fk 3694 2677 a(f) p Fn(z) p Fo 3826 2677 a(+) p Fn 0 2797 a(w) p Fo 73 2797 a(;) p Fn 117 2797 a(z) p Fk 205 2797 a(2) p Fn 311 2797 a(U) p Fk 387 2797 a(g) p Ff 479 2797 a(for) p 641 2797 a(some) p 896 2797 a(c) p 936 2797 a(onstant) p 1291 2797 a(ve) p 1376 2797 a(ctor) p Fn 1585 2797 a(w) p Fk 1697 2797 a(2) p Fj 1803 2797 a(R) p Fl 1890 2755 a(2) p Ff 1930 2797 a(.) p 2023 2797 a(L) p 2079 2797 a(et) p Fn 2198 2797 a(N) p Fo 2326 2797 a(=) p 2441 2797 a(0) p Fn(;) p Fo 2534 2797 a(1) p Fn(;) p Fo 2627 2797 a(2) p Fn(;) p 2720 2797 a(:) p 2764 2797 a(:) p 2808 2797 a(:) p Ff 2874 2797 a(or) p Fn 3006 2797 a(N) p Fo 3134 2797 a(=) p Fk 3250 2797 a(1) p Ff(,) p Fk 3422 2797 a(f) p Fn(z) p Fm 3517 2812 a(j) p Fk 3554 2797 a(g) p Fm 3604 2761 a(N) 3604 2822 y(j) p Fl 3637 2822 a(=1) p Ff 3771 2797 a(ar) p 3857 2797 a(e) 0 2918 y(\(at) p 169 2918 a(most\)) p 458 2918 a(c) p 498 2918 a(ountable) p 898 2918 a(p) p 943 2918 a(oints) p 1197 2918 a(in) p Fn 1329 2918 a(U) p Ff 1452 2918 a(with) p 1676 2918 a(no) p 1828 2918 a(ac) p 1918 2918 a(cumulation) p 2435 2918 a(p) p 2480 2918 a(oints) p 2734 2918 a(in) p Fn 2866 2918 a(U) p Ff 2942 2918 a(,) p Fn 3022 2918 a(S) p Fm 3082 2933 a(N) p Fo 3200 2918 a(=) p Fk 3326 2918 a([) p Fm 3392 2882 a(N) 3392 2942 y(j) p Fl 3425 2942 a(=1) p Fk 3519 2918 a(f) p Fn(z) p Fm 3614 2933 a(j) p Fk 3651 2918 a(g) p Ff 3748 2918 a(and) p Fn 0 3048 a(T) p Fm 57 3063 a(N) p Fo 164 3048 a(=) p Fn 279 3048 a(S) p Fm 339 3063 a(N) p Fo 433 3048 a(+) p Fn 535 3048 a(w) p Ff 649 3048 a(\() p Fn(S) p Fm 749 3063 a(N) p Fo 855 3048 a(=) p Fn 970 3048 a(T) p Fm 1027 3063 a(N) p Fo 1133 3048 a(=) p Fk 1248 3048 a(;) p Ff 1339 3048 a(when) p Fn 1595 3048 a(N) p Fo 1722 3048 a(=) p 1837 3048 a(0) p Ff(\).) p 2019 3048 a(L) p 2075 3048 a(et) p Fj 2194 3048 a(a) p Fk 2296 3048 a(2) p Fn 2401 3048 a(C) p Fg 2478 3012 a(1) p Fo 2553 3048 a(\() p Fn(U) p Fk 2694 3048 a(n) p Fn 2771 3048 a(S) p Fm 2831 3063 a(N) p Fo 2898 3048 a(;) p Fj 2942 3048 a(R) p Fl 3029 3006 a(2) p Fo 3069 3048 a(\)) p Fk 3133 3048 a(\\) p Fn 3226 3048 a(L) p Fl 3292 3012 a(1) p Fm 3292 3073 a(l) q(oc) p Fo 3384 3048 a(\() p Fn(U) p Fo 3498 3048 a(;) p Fj 3542 3048 a(R) p Fl 3629 3006 a(2) p Fo 3669 3048 a(\)) p Ff 3748 3048 a(and) p Fj 0 3169 a(b) p Fk 80 3169 a(2) p Fn 174 3169 a(C) p Fg 251 3133 a(1) p Fo 325 3169 a(\() p Fn(V) p Fk 464 3169 a(n) p Fn 536 3169 a(T) p Fm 593 3184 a(N) p Fo 660 3169 a(;) p Fj 704 3169 a(R) p Fl 791 3127 a(2) p Fo 831 3169 a(\)) p Fk 891 3169 a(\\) p Fn 979 3169 a(L) p Fl 1045 3133 a(1) p Fm 1045 3193 a(l) q(oc) p Fo 1136 3169 a(\() p Fn(V) p Fo 1253 3169 a(;) p Fj 1297 3169 a(R) p Fl 1384 3127 a(2) p Fo 1423 3169 a(\)) p Ff(.) p 1536 3169 a(Assume) p 1900 3169 a(that) p Fo 841 3440 a(rot) p Fj 982 3440 a(b) p Fo(\() p Fn(z) p Fo 1144 3440 a(+) p Fn 1242 3440 a(w) p Fo 1315 3440 a(\)) p 1380 3440 a(=) p 1483 3440 a(rot) p Fj 1624 3440 a(a) p Fo 1688 3440 a(\() p Fn(z) p Fo 1775 3440 a(\)) p 1835 3440 a(+) p Fm 1963 3332 a(N) p Fi 1935 3357 a(X) p Fm 1933 3539 a(j) p Fl 1966 3539 a(=1) p Fo 2072 3440 a(2) p Fn(\031) t(\013) p Fm 2242 3455 a(j) p Fn 2279 3440 a(\016) p Fo 2326 3440 a(\() p Fn(z) p Fk 2435 3440 a(\000) p Fn 2535 3440 a(z) p Fm 2580 3455 a(j) p Fo 2617 3440 a(\)) p 2690 3440 a(in) p Fk 2806 3440 a(D) p Fg 2886 3399 a(0) p Fo 2909 3440 a(\() p Fn(U) p Fo 3023 3440 a(\)) p 3701 3440 a(\(2.7\)) p Ff 0 3719 a(and) p 181 3719 a(that) p 371 3719 a(al) p 451 3719 a(l) p Fn 502 3719 a(\013) p Fm 564 3734 a(j) p Ff 601 3719 a('s) p 697 3719 a(ar) p 783 3719 a(e) p 854 3719 a(inte) p 1011 3719 a(gers.) p 1253 3719 a(Then) p 1499 3719 a(ther) p 1662 3719 a(e) p 1733 3719 a(exists) p 1991 3719 a(an) p 2122 3719 a(op) p 2217 3719 a(er) p 2298 3719 a(ator) p Fn 2497 3719 a(t) p Fg 2532 3734 a(\000) p Fm(w) p Ff 2669 3719 a(fr) p 2735 3719 a(om) p Fk 2891 3719 a(D) p Fg 2971 3683 a(0) p Fo 2994 3719 a(\() p Fn(V) p Fk 3114 3719 a(n) p Fn 3167 3719 a(T) p Fm 3224 3734 a(N) p Fo 3291 3719 a(\)) p Ff 3355 3719 a(to) p Fk 3464 3719 a(D) p Fg 3544 3683 a(0) p Fo 3566 3719 a(\() p Fn(U) p Fk 3683 3719 a(n) p Fn 3736 3719 a(S) p Fm 3796 3734 a(N) p Fo 3864 3719 a(\)) p Ff 0 3840 a(satisfying) p 433 3840 a(the) p 595 3840 a(fol) p 705 3840 a(lowing) p 1009 3840 a(c) p 1049 3840 a(onditions:) 146 3960 y(\(i\)) p 291 3960 a(Ther) p 492 3960 a(e) p 571 3960 a(exists) p Fo 837 3960 a(\010) p Fk 936 3960 a(2) p Fn 1030 3960 a(C) p Fg 1107 3924 a(1) p Fo 1181 3960 a(\() p Fn(U) p Fk 1318 3960 a(n) p Fn 1390 3960 a(S) p Fm 1450 3975 a(N) p Fo 1517 3960 a(\)) p Ff 1590 3960 a(with) p Fk 1802 3960 a(j) p Fo(\010\() p Fn(z) p Fo 1987 3960 a(\)) p Fk(j) p Fo 2081 3960 a(=) p 2184 3960 a(1) p Ff 2268 3960 a(for) p 2424 3960 a(every) p Fn 2682 3960 a(z) p Fk 2759 3960 a(2) p Fn 2853 3960 a(U) p Fk 2952 3960 a(n) p Fn 3024 3960 a(S) p Fm 3084 3975 a(N) p Ff 3186 3960 a(such) p 3408 3960 a(that) p Fn 1437 4157 a(t) p Fg 1472 4172 a(\000) p Fm(w) p Fn 1583 4157 a(u) p Fo(\() p Fn(z) p Fo 1726 4157 a(\)) p 1792 4157 a(=) p 1896 4157 a(\010\() p Fn(z) p Fo 2053 4157 a(\)) p Fn(u) p Fo(\() p Fn(z) p Fo 2257 4157 a(+) p Fn 2355 4157 a(w) p Fo 2428 4157 a(\)) p Ff 0 4353 a(for) p 156 4353 a(every) p Fn 413 4353 a(u) p Fk 497 4353 a(2) p 591 4353 a(D) p Fg 671 4317 a(0) p Fo 694 4353 a(\() p Fn(V) p Fk 832 4353 a(n) p Fn 904 4353 a(T) p Fm 961 4368 a(N) p Fo 1029 4353 a(\)) p Ff(.) 146 4473 y(\(ii\)) p 321 4473 a(The) p 520 4473 a(intertwining) p 1069 4473 a(pr) p 1155 4473 a(op) p 1250 4473 a(erty) p Fi 853 4610 a(\022) p Fo 924 4663 a(1) p 924 4708 49 4 v Fn 932 4799 a(i) p Fk 983 4731 a(r) p Fo 1088 4731 a(+) p Fj 1186 4731 a(a) p Fi 1249 4610 a(\023) p Fl 1311 4626 a(2) p Fn 1367 4731 a(t) p Fg 1402 4746 a(\000) p Fm(w) p Fn 1513 4731 a(u) p Fo 1596 4731 a(=) p Fn 1700 4731 a(t) p Fg 1735 4746 a(\000) p Fm(w) p Fi 1863 4610 a(\022) p Fo 1934 4663 a(1) p 1934 4708 V Fn 1942 4799 a(i) p Fk 1993 4731 a(r) p Fo 2098 4731 a(+) p Fj 2196 4731 a(b) p Fi 2248 4610 a(\023) p Fl 2309 4626 a(2) p Fn 2365 4731 a(u) p Fo 2456 4731 a(in) p Fk 2572 4731 a(D) p Fg 2652 4690 a(0) p Fo 2675 4731 a(\() p Fn(U) p Fk 2812 4731 a(n) p Fn 2883 4731 a(S) p Fm 2943 4746 a(N) p Fo 3011 4731 a(\)) p 3701 4731 a(\(2.8\)) p Ff 0 4965 a(holds) p 249 4965 a(for) p 405 4965 a(any) p Fn 592 4965 a(u) p Fk 675 4965 a(2) p Fn 769 4965 a(D) p Fg 853 4929 a(0) p Fo 876 4965 a(\() p Fn(V) p Fk 1014 4965 a(n) p Fn 1086 4965 a(T) p Fm 1143 4980 a(N) p Fo 1211 4965 a(\)) p Ff(.) p Fo 1902 5214 a(13) p eop %%Page: 14 14 14 13 bop Ff 146 631 a(Pr) p 248 631 a(o) p 293 631 a(of.) p Fo 451 631 a(Com) m(bining) p 962 631 a(Lemma) p 1326 631 a(2.2) p 1498 631 a(with) p 1736 631 a(Lemma) p 2100 631 a(2.3,) p 2303 631 a(w) m(e) p 2462 631 a(can) p 2657 631 a(pro) m(v) m(e) p 2935 631 a(that) p 3162 631 a(there) p 3426 631 a(exists) p 3711 631 a(\010) p Fk 3836 631 a(2) p Fn 0 751 a(C) p Fg 77 715 a(1) p Fo 152 751 a(\() p Fn(U) p Fk 288 751 a(n) p Fn 360 751 a(S) p Fm 420 766 a(N) p Fo 488 751 a(\)) p 558 751 a(with) p Fk 780 751 a(j) p Fo(\010) p Fk(j) p Fo 934 751 a(=) p 1037 751 a(1) p 1119 751 a(satisfying) 800 968 y(\010\() p Fn(z) p Fo 957 968 a(\)) p Fg 995 927 a(\000) p Fl(1) p Fk 1091 968 a(r) p Fo(\010\() p Fn(z) p Fo 1331 968 a(\)) p 1398 968 a(=) p Fk 1501 968 a(r) p Fo 1606 968 a(+) p Fn 1704 968 a(i) p Fj(b) p Fo(\() p Fn(z) p Fo 1899 968 a(+) p Fn 1997 968 a(w) p Fo 2070 968 a(\)) p Fk 2130 968 a(\000) p Fn 2229 968 a(i) p Fj(a) p Fo 2326 968 a(\() p Fn(z) p Fo 2413 968 a(\)) p 2484 968 a(in) p Fk 2598 968 a(D) p Fg 2678 927 a(0) p Fo 2701 968 a(\() p Fn(U) p Fk 2837 968 a(n) p Fn 2909 968 a(S) p Fm 2969 983 a(N) p Fo 3037 968 a(\)) p Fn(:) p Fo 0 1185 a(Th) m(us) p 247 1185 a(w) m(e) p 391 1185 a(obtain) 982 1401 y(\010\() p Fn(z) p Fo 1139 1401 a(\)) p Fg 1177 1360 a(\000) p Fl(1) p Fo 1272 1401 a(\() p Fk(r) p Fo 1415 1401 a(+) p Fn 1513 1401 a(i) p Fj(a) p Fo(\() p Fn(z) p Fo 1696 1401 a(\)\)\010\() p Fn(z) p Fo 1929 1401 a(\)) p 2052 1401 a(=) p Fk 2211 1401 a(r) p Fo 2316 1401 a(+) p Fn 2414 1401 a(i) p Fj(b) p Fo(\() p Fn(z) p Fo 2609 1401 a(+) p Fn 2707 1401 a(w) p Fo 2780 1401 a(\)) p Fn(;) p Fo 942 1547 a(\010\() p Fn(z) p Fo 1099 1547 a(\)) p Fg 1137 1505 a(\000) p Fl(1) p Fo 1233 1547 a(\() p Fk(r) p Fo 1376 1547 a(+) p Fn 1474 1547 a(i) p Fj(a) p Fo(\() p Fn(z) p Fo 1657 1547 a(\)\)) p Fl 1733 1505 a(2) p Fo 1773 1547 a(\010\() p Fn(z) p Fo 1930 1547 a(\)) p 2052 1547 a(=) p 2211 1547 a(\() p Fk(r) p Fo 2354 1547 a(+) p Fn 2452 1547 a(i) p Fj(b) p Fo(\() p Fn(z) p Fo 2647 1547 a(+) p Fn 2745 1547 a(w) p Fo 2818 1547 a(\)\)) p Fl 2894 1505 a(2) p Fn 2933 1547 a(:) p Fo 3701 1547 a(\(2.9\)) 0 1763 y(Put) p Fn 191 1763 a(t) p Fg 226 1778 a(\000) p Fm(w) p Fn 337 1763 a(v) p Fo 388 1763 a(\() p Fn(z) p Fo 475 1763 a(\)) p 541 1763 a(=) p 645 1763 a(\010\() p Fn(z) p Fo 802 1763 a(\)) p Fn(v) p Fo 891 1763 a(\() p Fn(z) p Fo 1001 1763 a(+) p Fn 1098 1763 a(w) p Fo 1171 1763 a(\)) p 1241 1763 a(for) p Fn 1390 1763 a(v) p Fk 1468 1763 a(2) p 1562 1763 a(D) p Fg 1642 1727 a(0) p Fo 1665 1763 a(\() p Fn(V) p Fk 1803 1763 a(n) p Fn 1874 1763 a(T) p Fm 1931 1778 a(N) p Fo 1999 1763 a(\)) p Fn(:) p Fo 2096 1763 a(One) p 2302 1763 a(can) p 2480 1763 a(easily) p 2749 1763 a(v) m(erify) p 3019 1763 a(that) p 3230 1763 a(\(2.8\)) p 3463 1763 a(holds) p 3718 1763 a(b) m(y) p 3853 1763 a(a) 0 1884 y(short) p 250 1884 a(computation) p 819 1884 a(using) p 1074 1884 a(\(2.9\).) p 1345 1816 64 4 v 1345 1909 4 93 v 1405 1909 V 1345 1912 64 4 v Fh 0 2216 a(3) p 242 2216 a(Single) p 719 2216 a(solenoid) p 1341 2216 a(op) t(erator) p Fo 0 2435 a(In) p 131 2435 a(this) p 330 2435 a(section,) p 694 2435 a(w) m(e) p 847 2435 a(shall) p 1084 2435 a(giv) m(e) p 1294 2435 a(some) p 1548 2435 a(prop) s(erties) p 2015 2435 a(of) p 2135 2435 a(the) p 2312 2435 a(single) p 2593 2435 a(solenoid) p 2977 2435 a(op) s(erator) p Fn 3379 2435 a(H) p Fl 3460 2450 a(1) p Fo 3499 2435 a(,) p 3570 2435 a(some) p 3823 2435 a(of) 0 2555 y(whic) m(h) p 279 2555 a(are) p 442 2555 a(giv) m(en) p 696 2555 a(in) p 810 2555 a([Ex-St-Vy].) 146 2676 y(W) p 238 2676 a(e) p 314 2676 a(shall) p 542 2676 a(in) m(tro) s(duce) p 976 2676 a(some) p 1220 2676 a(op) s(erators) p 1652 2676 a(and) p 1841 2676 a(functions) p 2262 2676 a(con) m(v) m(enien) m(t) p 2744 2676 a(for) p 2893 2676 a(later) p 3121 2676 a(calculus.) p Fq 146 2876 a(Prop) s(osition) p 752 2876 a(3.1) p Ff 944 2876 a(\(i\)) p 1095 2876 a(De\014ne) p 1410 2876 a(di\013er) p 1631 2876 a(ential) p 1909 2876 a(op) p 2004 2876 a(er) p 2085 2876 a(ators) p Fo 2339 2876 a(\005) p Fm 2412 2891 a(x) p Fn 2456 2876 a(;) p Fo 2500 2876 a(\005) p Fm 2573 2891 a(y) p Fn 2615 2876 a(;) p Fk 2659 2876 a(A) p Fl 2739 2891 a(1) p Fn 2777 2876 a(;) p Fk 2821 2876 a(A) p Fg 2901 2829 a(y) p Fl 2901 2898 a(1) p Ff 2982 2876 a(fr) p 3048 2876 a(om) p Fk 3219 2876 a(D) p Fg 3299 2840 a(0) p Fo 3322 2876 a(\() p Fj(R) p Fl 3448 2834 a(2) p Fk 3514 2876 a(n) p 3592 2876 a(f) p Fo(0) p Fk(g) p Fo(\)) p Ff 3820 2876 a(to) p Fk 0 2997 a(D) p Fg 80 2960 a(0) p Fo 103 2997 a(\() p Fj(R) p Fl 228 2955 a(2) p Fk 290 2997 a(n) p 362 2997 a(f) p Fo(0) p Fk(g) p Fo(\)) p Ff 583 2997 a(by) p Fo 1118 3262 a(\005) p Fm 1191 3277 a(x) p Fo 1318 3262 a(=) 1486 3194 y(1) p 1486 3238 49 4 v Fn 1494 3330 a(i) 1545 3262 y(@) p Fm 1596 3277 a(x) p Fk 1663 3262 a(\000) p Fn 1772 3194 a(B) p 1772 3238 80 4 v Fo 1787 3330 a(2) p Fn 1861 3262 a(y) p Fk 1935 3262 a(\000) p Fn 2085 3194 a(\013) p 2044 3238 145 4 v Fk 2044 3330 a(j) p Fn(z) p Fk 2121 3330 a(j) p Fl 2149 3301 a(2) p Fn 2199 3262 a(y) t(;) p Fo 1120 3523 a(\005) p Fm 1193 3538 a(y) p Fo 1318 3523 a(=) 1486 3456 y(1) p 1486 3500 49 4 v Fn 1494 3592 a(i) 1545 3523 y(@) p Fm 1596 3538 a(y) p Fo 1660 3523 a(+) p Fn 1768 3456 a(B) p 1768 3500 80 4 v Fo 1783 3592 a(2) p Fn 1857 3523 a(x) p Fo 1935 3523 a(+) p Fn 2084 3456 a(\013) p 2043 3500 145 4 v Fk 2043 3592 a(j) p Fn(z) p Fk 2120 3592 a(j) p Fl 2148 3563 a(2) p Fn 2197 3523 a(x;) p Fk 1116 3785 a(A) p Fl 1196 3800 a(1) p Fo 1318 3785 a(=) p Fn 1476 3785 a(i) p Fo(\005) p Fm 1582 3800 a(x) p Fo 1649 3785 a(+) p 1747 3785 a(\005) p Fm 1820 3800 a(y) p Fo 1889 3785 a(=) p 1993 3785 a(2) p Fn(@) p Fm 2093 3800 a(z) p Fo 2155 3785 a(+) p Fn 2263 3718 a(B) p 2263 3762 80 4 v Fo 2278 3854 a(2) 2358 3785 y(\026) p Fn 2352 3785 a(z) p Fo 2424 3785 a(+) p Fn 2532 3718 a(\013) p 2532 3762 63 4 v 2538 3854 a(z) p Fk 1116 4022 a(A) p Fg 1196 3975 a(y) p Fl 1196 4044 a(1) p Fo 1318 4022 a(=) p Fk 1476 4022 a(\000) p Fn(i) p Fo(\005) p Fm 1659 4037 a(x) p Fo 1726 4022 a(+) p 1824 4022 a(\005) p Fm 1897 4037 a(y) p Fo 1967 4022 a(=) p Fk 2070 4022 a(\000) p Fo(2) p Fn(@) p Fl 2251 4037 a(\026) p Fm 2247 4037 a(z) p Fo 2310 4022 a(+) p Fn 2418 3955 a(B) p 2418 3999 80 4 v Fo 2433 4091 a(2) p Fn 2507 4022 a(z) p Fo 2579 4022 a(+) p Fn 2687 3955 a(\013) p 2687 3999 63 4 v Fo 2699 4091 a(\026) p Fn 2693 4091 a(z) 2759 4022 y(;) p Ff 0 4265 a(wher) p 196 4265 a(e) p Fn 1189 4404 a(@) p Fm 1240 4419 a(z) p Fo 1308 4404 a(=) 1421 4337 y(1) p 1421 4381 49 4 v 1421 4473 a(2) 1480 4404 y(\() p Fn(@) p Fm 1569 4419 a(x) p Fk 1635 4404 a(\000) p Fn 1735 4404 a(i@) p Fm 1819 4419 a(y) p Fo 1861 4404 a(\)) p Fn(;) p 1978 4404 a(@) p Fl 2033 4419 a(\026) p Fm 2029 4419 a(z) p Fo 2096 4404 a(=) 2210 4337 y(1) p 2210 4381 V 2210 4473 a(2) 2269 4404 y(\() p Fn(@) p Fm 2358 4419 a(x) p Fo 2424 4404 a(+) p Fn 2522 4404 a(i@) p Fm 2606 4419 a(y) p Fo 2648 4404 a(\)) p Fn(:) p Ff 0 4603 a(Then,) p 284 4603 a(the) p 446 4603 a(fol) p 556 4603 a(lowing) p 859 4603 a(r) p 895 4603 a(elation) p 1217 4603 a(holds:) p Fk 905 4819 a(A) p Fg 985 4772 a(y) p Fl 985 4841 a(1) p Fk 1024 4819 a(A) p Fl 1104 4834 a(1) p Fo 1171 4819 a(=) p Fk 1274 4819 a(L) p Fl 1343 4834 a(1) p Fk 1405 4819 a(\000) p Fn 1504 4819 a(B) p 1583 4819 a(;) p Fk 1828 4819 a(A) p Fl 1908 4834 a(1) p Fk 1947 4819 a(A) p Fg 2027 4772 a(y) p Fl 2027 4841 a(1) p Fo 2094 4819 a(=) p Fk 2197 4819 a(L) p Fl 2266 4834 a(1) p Fo 2328 4819 a(+) p Fn 2426 4819 a(B) p 2505 4819 a(;) p Fo 841 4965 a(\() p Fk(A) p Fl 959 4980 a(1) p Fn 998 4965 a(u;) p 1098 4965 a(v) p Fo 1149 4965 a(\)) p 1214 4965 a(=) p 1317 4965 a(\() p Fn(u;) p Fk 1455 4965 a(A) p Fg 1535 4917 a(y) p Fl 1535 4986 a(1) p Fn 1573 4965 a(v) p Fo 1624 4965 a(\)) p 1828 4965 a(for) p 1977 4965 a(an) m(y) p Fn 2165 4965 a(u;) p 2265 4965 a(v) p Fk 2343 4965 a(2) p Fn 2437 4965 a(C) p Fg 2514 4924 a(1) p Fl 2507 4989 a(0) p Fo 2589 4965 a(\() p Fj(R) p Fl 2714 4923 a(2) p Fk 2775 4965 a(n) p 2847 4965 a(f) p Fo(0) p Fk(g) p Fo(\)) p Fn(:) p Fo 1902 5214 a(14) p eop %%Page: 15 15 15 14 bop Ff 146 631 a(\(ii\)) p 321 631 a(De\014ne) p 628 631 a(smo) p 793 631 a(oth) p 960 631 a(functions) p Fk 1383 631 a(f) p Fn(\036) p Fm 1491 646 a(m;n) p Fk 1620 631 a(g) p Ff(,) p Fk 1734 631 a(f) p Fn( ) p Fm 1847 646 a(m;n) p Fk 1976 631 a(g) p Fo 2061 631 a(\() p Fn(m;) p 2228 631 a(n) p Fk 2314 631 a(2) p Fj 2408 631 a(Z) p Fo 2492 631 a(\)) p Ff 2565 631 a(on) p Fj 2704 631 a(R) p Fl 2792 589 a(2) p Fk 2853 631 a(n) p 2925 631 a(f) p Fo(0) p Fk(g) p Ff 3108 631 a(by) p Fn 1311 846 a(\036) p Fm 1369 861 a(m;n) p Fo 1498 846 a(\() p Fn(z) p Fo 1585 846 a(\)) p 1706 846 a(=) p Fk 1865 846 a(j) p Fn(z) p Fk 1942 846 a(j) p Fm 1970 805 a(\013) p Fn 2020 846 a(z) p Fm 2069 805 a(m) p Fo 2142 846 a(\026) p Fn 2136 846 a(z) p Fm 2185 805 a(n) p Fn 2233 846 a(e) p Fg 2278 805 a(\000) p Fe 2343 778 a(B) p 2343 790 49 4 v Fd 2352 831 a(4) p Fg 2401 805 a(j) p Fm(z) p Fg 2457 805 a(j) p Fd 2477 781 a(2) p Fn 2515 846 a(;) 1305 1009 y( ) p Fm 1368 1024 a(m;n) p Fo 1498 1009 a(\() p Fn(z) p Fo 1585 1009 a(\)) p 1706 1009 a(=) p Fk 1865 1009 a(j) p Fn(z) p Fk 1942 1009 a(j) p Fg 1970 968 a(\000) p Fm(\013) p Fn 2075 1009 a(z) p Fm 2124 968 a(m) p Fo 2197 1009 a(\026) p Fn 2191 1009 a(z) p Fm 2240 968 a(n) p Fn 2288 1009 a(e) p Fg 2333 968 a(\000) p Fe 2398 941 a(B) p 2398 953 V Fd 2407 994 a(4) p Fg 2456 968 a(j) p Fm(z) p Fg 2512 968 a(j) p Fd 2532 945 a(2) p Fn 2570 1009 a(:) p Ff 0 1206 a(Then) p 254 1206 a(the) p 416 1206 a(fol) p 526 1206 a(lowing) p 829 1206 a(r) p 865 1206 a(elation) p 1187 1206 a(holds) p 1436 1206 a(for) p Fn 1592 1206 a(m;) p 1721 1206 a(n) p Fk 1806 1206 a(2) p Fj 1901 1206 a(Z) p Ff 1984 1206 a(:) p Fk 1092 1403 a(A) p Fl 1172 1418 a(1) p Fn 1211 1403 a(\036) p Fm 1269 1418 a(m;n) p Fo 1481 1403 a(=) p 1640 1403 a(2\() p Fn(\013) p Fo 1811 1403 a(+) p Fn 1909 1403 a(m) p Fo(\)) p Fn(\036) p Fm 2090 1418 a(m) p Fg(\000) p Fl(1) p Fm(;n) p Fn 2310 1403 a(;) p Fk 1092 1558 a(A) p Fg 1172 1511 a(y) p Fl 1172 1580 a(1) p Fn 1211 1558 a(\036) p Fm 1269 1573 a(m;n) p Fo 1481 1558 a(=) p Fk 1640 1558 a(\000) p Fo(2) p Fn(n\036) p Fm 1882 1573 a(m;n) p Fg(\000) p Fl(1) p Fo 2123 1558 a(+) p Fn 2221 1558 a(B) p 2300 1558 a(\036) p Fm 2358 1573 a(m) p Fl(+1) p Fm(;n) p Fn 2577 1558 a(;) p Fk 1087 1703 a(A) p Fl 1167 1718 a(1) p Fn 1206 1703 a( ) p Fm 1269 1718 a(m;n) p Fo 1481 1703 a(=) p 1640 1703 a(2) p Fn(m ) p Fm 1837 1718 a(m) p Fg(\000) p Fl(1) p Fm(;n) p Fn 2057 1703 a(;) p Fk 1087 1858 a(A) p Fg 1167 1810 a(y) p Fl 1167 1879 a(1) p Fn 1206 1858 a( ) p Fm 1269 1873 a(m;n) p Fo 1481 1858 a(=) p 1640 1858 a(2\() p Fn(\013) p Fk 1811 1858 a(\000) p Fn 1911 1858 a(n) p Fo(\)) p Fn( ) p Fm 2070 1873 a(m;n) p Fg(\000) p Fl(1) p Fo 2312 1858 a(+) p Fn 2410 1858 a(B) p 2489 1858 a( ) p Fm 2552 1873 a(m) p Fl(+1) p Fm(;n) p Fn 2772 1858 a(;) p Fk 1111 2003 a(L) p Fl 1180 2018 a(1) p Fn 1219 2003 a(\036) p Fm 1277 2018 a(m;) p Fl(0) p Fo 1481 2003 a(=) p 1640 2003 a(\(2) p Fn(\013) p Fo 1811 2003 a(+) p 1909 2003 a(2) p Fn(m) p Fo 2066 2003 a(+) p 2164 2003 a(1\)) p Fn(B) p 2330 2003 a(\036) p Fm 2388 2018 a(m;) p Fl(0) p Fn 2509 2003 a(;) p Fk 1125 2148 a(L) p Fl 1194 2163 a(1) p Fn 1233 2148 a( ) p Fl 1296 2163 a(0) p Fm(;n) p Fo 1481 2148 a(=) p Fn 1640 2148 a(B) p 1719 2148 a( ) p Fl 1782 2163 a(0) p Fm(;n) p Fn 1884 2148 a(:) p Ff 0 2345 a(In) p 127 2345 a(p) p 172 2345 a(articular,) p 603 2345 a(the) p 765 2345 a(four) p 973 2345 a(functions) p Fn 1006 2542 a(\036) p Fg 1064 2557 a(\000) p Fl(1) p Fo 1186 2542 a(=) p Fn 1289 2542 a(\036) p Fg 1347 2557 a(\000) p Fl(1) p Fm(;) p Fl(0) p Fn 1496 2542 a(;) p 1575 2542 a( ) p Fl 1638 2557 a(1) p Fo 1705 2542 a(=) p Fn 1809 2542 a( ) p Fl 1872 2557 a(0) p Fm(;) p Fl(1) p Fn 1967 2542 a(;) p 2045 2542 a(\036) p Fl 2103 2557 a(0) p Fo 2170 2542 a(=) p Fn 2274 2542 a(\036) p Fl 2332 2557 a(0) p Fm(;) p Fl(0) p Fn 2426 2542 a(;) p 2504 2542 a( ) p Fl 2567 2557 a(0) p Fo 2635 2542 a(=) p Fn 2738 2542 a( ) p Fl 2801 2557 a(0) p Fm(;) p Fl(0) p Fo 3701 2542 a(\(3.1\)) p Ff 0 2739 a(satisfy) p Fk 1080 2936 a(L) p Fl 1149 2951 a(1) p Fn 1188 2936 a(\036) p Fg 1246 2951 a(\000) p Fl(1) p Fo 1368 2936 a(=) p 1472 2936 a(\(2) p Fn(\013) p Fk 1643 2936 a(\000) p Fo 1743 2936 a(1\)) p Fn(B) p 1909 2936 a(\036) p Fg 1967 2951 a(\000) p Fl(1) p Fn 2061 2936 a(;) p Fk 2254 2936 a(L) p Fl 2323 2951 a(1) p Fn 2362 2936 a( ) p Fl 2425 2951 a(1) p Fo 2493 2936 a(=) p Fn 2596 2936 a(B) p 2675 2936 a( ) p Fl 2738 2951 a(1) p Fn 2778 2936 a(;) p Fk 1192 3081 a(L) p Fl 1261 3096 a(1) p Fn 1300 3081 a(\036) p Fl 1358 3096 a(0) p Fo 1425 3081 a(=) p 1528 3081 a(\(2) p Fn(\013) p Fo 1700 3081 a(+) p 1798 3081 a(1\)) p Fn(B) p 1964 3081 a(\036) p Fl 2022 3096 a(0) p Fn 2061 3081 a(;) p Fk 2254 3081 a(L) p Fl 2323 3096 a(1) p Fn 2362 3081 a( ) p Fl 2425 3096 a(0) p Fo 2493 3081 a(=) p Fn 2596 3081 a(B) p 2675 3081 a( ) p Fl 2738 3096 a(0) p Fn 2778 3081 a(:) p Fo 3701 3081 a(\(3.2\)) p Ff 146 3283 a(Pr) p 248 3283 a(o) p 293 3283 a(of.) p Fo 435 3283 a(W) p 527 3283 a(e) p 603 3283 a(can) p 782 3283 a(pro) m(v) m(e) p 1045 3283 a(statemen) m(ts) p 1537 3283 a(ab) s(o) m(v) m(e) p 1813 3283 a(b) m(y) p 1949 3283 a(the) p 2117 3283 a(follo) m(wing) p 2528 3283 a(form) m(ulas.) 1287 3523 y([\005) p Fm 1387 3538 a(x) p Fn 1431 3523 a(;) p Fo 1475 3523 a(\005) p Fm 1548 3538 a(y) p Fo 1589 3523 a(]) p 1644 3523 a(=) p Fn 1757 3456 a(B) p 1757 3500 80 4 v 1780 3592 a(i) 1846 3523 y(;) p Fk 2040 3523 a(L) p Fl 2109 3538 a(1) p Fo 2175 3523 a(=) p 2279 3523 a(\005) p Fl 2352 3482 a(2) p Fm 2352 3548 a(x) p Fo 2418 3523 a(+) p 2516 3523 a(\005) p Fl 2589 3482 a(2) p Fm 2589 3548 a(y) p Fn 822 3735 a(@) p Fm 873 3750 a(z) p Fn 914 3735 a(z) p Fm 963 3694 a(m) p Fo 1058 3735 a(=) p Fn 1171 3668 a(m) p 1171 3712 86 4 v 1189 3803 a(z) 1266 3735 y(z) p Fm 1315 3694 a(m) p Fn 1383 3735 a(;) p 1459 3735 a(@) p Fl 1514 3750 a(\026) p Fm 1510 3750 a(z) p Fn 1550 3735 a(z) p Fm 1599 3694 a(m) p Fo 1694 3735 a(=) p 1798 3735 a(0) p Fn(;) p 2040 3735 a(@) p Fm 2091 3750 a(z) p Fo 2137 3735 a(\026) p Fn 2131 3735 a(z) p Fm 2180 3694 a(n) p Fo 2255 3735 a(=) p 2359 3735 a(0) p Fn(;) p 2484 3735 a(@) p Fl 2539 3750 a(\026) p Fm 2535 3750 a(z) p Fo 2581 3735 a(\026) p Fn 2575 3735 a(z) p Fm 2624 3694 a(n) p Fo 2699 3735 a(=) p Fn 2813 3668 a(n) p 2813 3712 59 4 v Fo 2823 3803 a(\026) p Fn 2817 3803 a(z) p Fo 2887 3735 a(\026) p Fn 2881 3735 a(z) p Fm 2930 3694 a(n) p Fn 2978 3735 a(;) 969 3962 y(@) p Fm 1020 3977 a(z) p Fo 1060 3962 a(\() p Fk(j) p Fn(z) p Fk 1175 3962 a(j) p Fg 1203 3921 a(\006) p Fm(\013) p Fo 1308 3962 a(\)) p 1373 3962 a(=) p Fk 1487 3895 a(\006) p Fn(\013) p 1487 3939 141 4 v Fo 1508 4030 a(2) p Fn(z) p Fk 1637 3962 a(j) p Fn(z) p Fk 1714 3962 a(j) p Fg 1742 3921 a(\006) p Fm(\013) p Fn 1846 3962 a(;) p 2040 3962 a(@) p Fl 2095 3977 a(\026) p Fm 2091 3977 a(z) p Fo 2131 3962 a(\() p Fk(j) p Fn(z) p Fk 2246 3962 a(j) p Fg 2274 3921 a(\006) p Fm(\013) p Fo 2378 3962 a(\)) p 2444 3962 a(=) p Fk 2557 3895 a(\006) p Fn(\013) p 2557 3939 V Fo 2578 4030 a(2) p 2633 4030 a(\026) p Fn 2627 4030 a(z) p Fk 2708 3962 a(j) p Fn(z) p Fk 2785 3962 a(j) p Fg 2813 3921 a(\006) p Fm(\013) p Fn 2917 3962 a(;) 833 4199 y(@) p Fm 884 4214 a(z) p Fn 925 4199 a(e) p Fg 970 4158 a(\000) p Fe 1035 4131 a(B) p 1035 4143 49 4 v Fd 1044 4184 a(4) p Fg 1093 4158 a(j) p Fm(z) p Fg 1149 4158 a(j) p Fd 1169 4135 a(2) p Fo 1234 4199 a(=) p Fk 1338 4199 a(\000) p Fn 1425 4132 a(B) p 1425 4176 80 4 v Fo 1440 4267 a(4) 1520 4199 y(\026) p Fn 1515 4199 a(z) t(e) p Fg 1609 4158 a(\000) p Fe 1674 4131 a(B) p 1674 4143 49 4 v Fd 1683 4184 a(4) p Fg 1733 4158 a(j) p Fm(z) p Fg 1789 4158 a(j) p Fd 1809 4135 a(2) p Fn 1846 4199 a(;) p 2040 4199 a(@) p Fl 2095 4214 a(\026) p Fm 2091 4214 a(z) p Fn 2131 4199 a(e) p Fg 2176 4158 a(\000) p Fe 2241 4131 a(B) p 2241 4143 V Fd 2250 4184 a(4) p Fg 2299 4158 a(j) p Fm(z) p Fg 2355 4158 a(j) p Fd 2375 4135 a(2) p Fo 2441 4199 a(=) p Fk 2544 4199 a(\000) p Fn 2631 4132 a(B) p 2631 4176 80 4 v Fo 2646 4267 a(4) p Fn 2721 4199 a(z) t(e) p Fg 2815 4158 a(\000) p Fe 2880 4131 a(B) p 2881 4143 49 4 v Fd 2890 4184 a(4) p Fg 2939 4158 a(j) p Fm(z) p Fg 2995 4158 a(j) p Fd 3015 4135 a(2) p Fn 3053 4199 a(:) p 0 4356 64 4 v 0 4448 4 93 v 60 4448 V 0 4451 64 4 v Fo 0 4625 a(Remark) p 367 4625 a(1.) p 486 4625 a(In) p 608 4625 a(the) p 776 4625 a(p) s(olar) p 1028 4625 a(co) s(ordinate) p Fn 1508 4625 a(z) p Fo 1585 4625 a(=) p Fn 1689 4625 a(r) s(e) p Fm 1781 4589 a(i\022) p Fo 1844 4625 a(,) p Fn 1248 4828 a(\036) p Fm 1306 4843 a(m;n) p Fo 1434 4828 a(\() p Fn(z) p Fo 1521 4828 a(\)) p 1588 4828 a(=) p Fn 1691 4828 a(r) p Fm 1738 4792 a(\013) p Fl(+) p Fm(m) p Fl(+) p Fm(n) p Fn 2002 4828 a(e) p Fm 2047 4792 a(i) p Fl(\() p Fm(m) p Fg(\000) p Fm(n) p Fl(\)) p Fm(\022) p Fn 2326 4828 a(e) p Fg 2371 4791 a(\000) p Fe 2436 4764 a(B) p 2436 4776 49 4 v Fd 2445 4817 a(4) p Fm 2494 4791 a(r) p Fd 2528 4768 a(2) p Fn 2567 4828 a(;) 1248 4962 y( ) p Fm 1311 4977 a(m;n) p Fo 1440 4962 a(\() p Fn(z) p Fo 1527 4962 a(\)) p 1593 4962 a(=) p Fn 1697 4962 a(r) p Fg 1744 4926 a(\000) p Fm(\013) p Fl(+) p Fm(m) p Fl(+) p Fm(n) p Fn 2063 4962 a(e) p Fm 2108 4926 a(i) p Fl(\() p Fm(m) p Fg(\000) p Fm(n) p Fl(\)) p Fm(\022) p Fn 2386 4962 a(e) p Fg 2431 4925 a(\000) p Fe 2496 4898 a(B) p 2496 4910 V Fd 2505 4951 a(4) p Fm 2555 4925 a(r) p Fd 2589 4902 a(2) p Fn 2627 4962 a(:) p Fo 3701 4889 a(\(3.3\)) 1902 5214 y(15) p eop %%Page: 16 16 16 15 bop Fo 0 631 a(This) p 223 631 a(expressions) p 731 631 a(imply) p 1005 631 a(that) p 1216 631 a(the) p 1384 631 a(four) p 1587 631 a(functions) p Fn 2008 631 a(\036) p Fg 2066 646 a(\000) p Fl(1) p Fo 2160 631 a(,) p Fn 2220 631 a( ) p Fl 2283 646 a(1) p Fo 2322 631 a(,) p Fn 2382 631 a(\036) p Fl 2440 646 a(0) p Fo 2479 631 a(,) p Fn 2539 631 a( ) p Fl 2602 646 a(0) p Fo 2674 631 a(b) s(elong) p 2986 631 a(to) p Fn 3105 631 a(L) p Fl 3171 595 a(2) p Fo 3211 631 a(\() p Fj(R) p Fl 3336 589 a(2) p Fo 3376 631 a(\).) 0 751 y(Remark) p 369 751 a(2.) p 494 751 a(When) p 780 751 a(w) m(e) p 926 751 a(need) p 1155 751 a(to) p 1276 751 a(indicate) p 1647 751 a(the) p 1816 751 a(v) p 1862 751 a(alue) p Fn 2070 751 a(\013) p Fo 2167 751 a(explicitly) p 2551 751 a(,) p 2611 751 a(w) m(e) p 2757 751 a(denote) p Fn 3073 751 a(\036) p Fm 3131 715 a(\013) p Fg 3131 776 a(\000) p Fl(1) p Fo 3225 751 a(,) p Fn 3287 751 a( ) p Fm 3354 715 a(\013) p Fl 3350 776 a(1) p Fo 3404 751 a(,) p Fn 3465 751 a(\036) p Fm 3523 715 a(\013) p Fl 3523 776 a(0) p Fo 3573 751 a(,) p Fn 3635 751 a( ) p Fm 3702 715 a(\013) p Fl 3698 776 a(0) p Fo 3785 751 a(for) p Fn 0 872 a(\036) p Fg 58 887 a(\000) p Fl(1) p Fo 152 872 a(,) p Fn 212 872 a( ) p Fl 275 887 a(1) p Fo 315 872 a(,) p Fn 374 872 a(\036) p Fl 432 887 a(0) p Fo 471 872 a(,) p Fn 531 872 a( ) p Fl 594 887 a(0) p Fo 634 872 a(.) 146 992 y(W) p 238 992 a(e) p 314 992 a(quote) p 583 992 a(a) p 664 992 a(part) p 876 992 a(of) p 987 992 a(the) p 1155 992 a(result) p 1426 992 a(b) m(y) p 1562 992 a(Exner,) 1878 967 y(\024) 1875 992 y(St'o) m(v) p 2080 992 a(\023) p 2091 992 a(\020) m(\024) p 2119 992 a(cek) p 2290 992 a(and) p 2479 992 a(Vyt) p 2636 992 a(\024) p 2642 992 a(ras) p 2800 992 a(\([Ex-St-Vy]\)) p 3362 992 a(for) p 3511 992 a(later) p 3739 992 a(use.) 0 1112 y(Notice) p 306 1112 a(that) p 518 1112 a(some) p 762 1112 a(notes) p 1017 1112 a(used) p 1240 1112 a(in) p 1354 1112 a([Ex-St-Vy]) p 1840 1112 a(are) p 2003 1112 a(c) m(hanged.) p Fq 146 1279 a(Prop) s(osition) p 752 1279 a(3.2) p 932 1279 a(\(Exner-) 1308 1255 y(\024) 1305 1279 y(St'o) m(v) p 1542 1279 a(\023) p 1554 1279 a(\020) m(\024) p 1586 1279 a(cek-Vyt) p 1966 1279 a(\024) p 1971 1279 a(ras\)) p Ff 2207 1279 a(\(i\)) p 2359 1279 a(The) p 2566 1279 a(domain) p 2923 1279 a(of) p 3045 1279 a(the) p 3214 1279 a(adjoint) p 3553 1279 a(op) p 3648 1279 a(er) p 3729 1279 a(ator) p Fn 0 1399 a(L) p Fg 66 1363 a(\003) p Fl 66 1424 a(1) p Ff 141 1399 a(of) p Fn 255 1399 a(L) p Fl 321 1414 a(1) p Ff 396 1399 a(is) p Fn 750 1532 a(D) p Fo 834 1532 a(\() p Fn(L) p Fg 938 1491 a(\003) p Fl 938 1557 a(1) p Fo 978 1532 a(\)) p 1043 1532 a(=) p Fi 1147 1436 a(n) p Fn 1202 1532 a(u) p Fk 1285 1532 a(2) p Fn 1379 1532 a(L) p Fl 1445 1491 a(2) p Fo 1485 1532 a(\() p Fj(R) p Fl 1610 1490 a(2) p Fo 1650 1532 a(\)) p Fk 1710 1532 a(\\) p Fn 1799 1532 a(H) p Fl 1888 1491 a(2) p Fm 1880 1557 a(l) q(oc) p Fo 1970 1532 a(\() p Fj(R) p Fl 2095 1490 a(2) p Fk 2157 1532 a(n) p 2229 1532 a(f) p Fo(0) p Fk(g) p Fo(\);) p Fk 2460 1532 a(L) p Fl 2529 1547 a(1) p Fn 2567 1532 a(u) p Fk 2650 1532 a(2) p Fn 2744 1532 a(L) p Fl 2810 1491 a(2) p Fo 2850 1532 a(\() p Fj(R) p Fl 2975 1490 a(2) p Fo 3015 1532 a(\)) p Fi 3053 1436 a(o) p Fn 3125 1532 a(:) p Ff 146 1700 a(\(ii\)) p 321 1700 a(The) p 520 1700 a(de\014ciency) p 971 1700 a(indic) p 1176 1700 a(es) p 1295 1700 a(of) p Fn 1409 1700 a(L) p Fl 1475 1715 a(1) p Ff 1550 1700 a(ar) p 1636 1700 a(e) p Fo 1715 1700 a(\(2) p Fn(;) p Fo 1846 1700 a(2\)) p Ff(.) 146 1821 y(\(iii\)) p 351 1821 a(The) p 550 1821 a(four) p 758 1821 a(line) p 908 1821 a(ar) p 1033 1821 a(functionals) p Fo 612 2044 a(\010) p Fg 682 2059 a(\000) p Fl(1) p Fo 777 2044 a(\() p Fn(u) p Fo(\)) p 991 2044 a(=) p 1179 2044 a(lim) p Fm 1150 2101 a(r) p Fg 1184 2101 a(!) p Fl(+0) p Fn 1361 2044 a(r) p Fl 1408 2003 a(1) p Fg(\000) p Fm(\013) p Fo 1587 1977 a(1) p 1557 2021 108 4 v 1557 2113 a(2) p Fn(\031) p Fi 1692 1927 a(Z) p Fl 1775 1954 a(2) p Fm(\031) p Fl 1738 2116 a(0) p Fn 1874 2044 a(u) p Fo(\() p Fn(r) s(e) p Fm 2060 2003 a(i\022) p Fo 2122 2044 a(\)) p Fj(e) p Fm 2215 2003 a(i\022) p Fn 2278 2044 a(d\022) s(;) p Fo 661 2283 a(\011) p Fl 737 2298 a(1) p Fo 777 2283 a(\() p Fn(u) p Fo(\)) p 991 2283 a(=) p 1179 2283 a(lim) p Fm 1150 2340 a(r) p Fg 1184 2340 a(!) p Fl(+0) p Fn 1361 2283 a(r) p Fg 1408 2242 a(\000) p Fl(1+) p Fm(\013) p Fi 1619 2162 a(\022) p Fo 1719 2216 a(1) p 1690 2260 V 1690 2351 a(2) p Fn(\031) p Fi 1824 2166 a(Z) p Fl 1907 2192 a(2) p Fm(\031) p Fl 1870 2355 a(0) p Fn 2006 2283 a(u) p Fo(\() p Fn(r) s(e) p Fm 2192 2242 a(i\022) p Fo 2255 2283 a(\)) p Fj(e) p Fm 2348 2242 a(i\022) p Fn 2411 2283 a(d\022) p Fk 2532 2283 a(\000) p Fo 2631 2283 a(\010) p Fg 2701 2298 a(\000) p Fl(1) p Fo 2796 2283 a(\() p Fn(u) p Fo(\)) p Fn(r) p Fg 2975 2242 a(\000) p Fl(1+) p Fm(\013) p Fi 3169 2162 a(\023) p Fn 3246 2283 a(;) p Fo 661 2525 a(\011) p Fl 737 2540 a(0) p Fo 777 2525 a(\() p Fn(u) p Fo(\)) p 991 2525 a(=) p 1179 2525 a(lim) p Fm 1150 2581 a(r) p Fg 1184 2581 a(!) p Fl(+0) p Fn 1361 2525 a(r) p Fm 1408 2484 a(\013) p Fo 1497 2457 a(1) p 1467 2501 V 1467 2593 a(2) p Fn(\031) p Fi 1601 2408 a(Z) p Fl 1684 2434 a(2) p Fm(\031) p Fl 1648 2596 a(0) p Fn 1783 2525 a(u) p Fo(\() p Fn(r) s(e) p Fm 1969 2484 a(i\022) p Fo 2032 2525 a(\)) p Fn(d\022) s(;) p Fo 667 2763 a(\010) p Fl 737 2778 a(0) p Fo 777 2763 a(\() p Fn(u) p Fo(\)) p 991 2763 a(=) p 1179 2763 a(lim) p Fm 1150 2820 a(r) p Fg 1184 2820 a(!) p Fl(+0) p Fn 1361 2763 a(r) p Fg 1408 2722 a(\000) p Fm(\013) p Fi 1529 2642 a(\022) p Fo 1629 2696 a(1) p 1600 2740 V 1600 2832 a(2) p Fn(\031) p Fi 1734 2646 a(Z) p Fl 1817 2673 a(2) p Fm(\031) p Fl 1780 2835 a(0) p Fn 1916 2763 a(u) p Fo(\() p Fn(r) s(e) p Fm 2102 2722 a(i\022) p Fo 2164 2763 a(\)) p Fn(d\022) p Fk 2323 2763 a(\000) p Fo 2423 2763 a(\011) p Fl 2499 2778 a(0) p Fo 2538 2763 a(\() p Fn(u) p Fo(\)) p Fn(r) p Fg 2717 2722 a(\000) p Fm(\013) p Fi 2821 2642 a(\023) p Ff 0 2978 a(ar) p 86 2978 a(e) p 166 2978 a(wel) p 306 2978 a(l-de\014ne) p 611 2978 a(d) p 694 2978 a(and) p 883 2978 a(\014nite) p 1135 2978 a(for) p Fn 1290 2978 a(u) p Fk 1373 2978 a(2) p Fn 1467 2978 a(D) p Fo 1551 2978 a(\() p Fn(L) p Fg 1655 2942 a(\003) p Fl 1655 3003 a(1) p Fo 1695 2978 a(\)) p Ff(.) 146 3098 y(\(iv\)) p 336 3098 a(Every) p Fn 615 3098 a(u) p Fk 698 3098 a(2) p Fn 792 3098 a(D) p Fo 876 3098 a(\() p Fn(L) p Fg 980 3062 a(\003) p Fl 980 3123 a(1) p Fo 1019 3098 a(\)) p Ff 1092 3098 a(is) p 1197 3098 a(uniquely) p 1583 3098 a(de) p 1673 3098 a(c) p 1713 3098 a(omp) p 1888 3098 a(ose) p 2018 3098 a(d) p 2101 3098 a(as) p Fn 837 3275 a(u) p Fo 920 3275 a(=) p 1024 3275 a(\010) p Fg 1094 3290 a(\000) p Fl(1) p Fo 1189 3275 a(\() p Fn(u) p Fo(\)) p Fn(\036) p Fg 1379 3290 a(\000) p Fl(1) p Fo 1494 3275 a(+) p 1592 3275 a(\011) p Fl 1668 3290 a(1) p Fo 1708 3275 a(\() p Fn(u) p Fo(\)) p Fn( ) p Fl 1903 3290 a(1) p Fo 1964 3275 a(+) p 2062 3275 a(\010) p Fl 2132 3290 a(0) p Fo 2172 3275 a(\() p Fn(u) p Fo(\)) p Fn(\036) p Fl 2362 3290 a(0) p Fo 2423 3275 a(+) p 2521 3275 a(\011) p Fl 2597 3290 a(0) p Fo 2636 3275 a(\() p Fn(u) p Fo(\)) p Fn( ) p Fl 2831 3290 a(0) p Fo 2892 3275 a(+) p Fn 2990 3275 a(\030) p 3038 3275 a(;) p Fo 3701 3275 a(\(3.4\)) p Ff 0 3452 a(wher) p 196 3452 a(e) p Fn 275 3452 a(\030) p Fk 350 3452 a(2) p Fn 444 3452 a(D) p Fo 528 3452 a(\() p 566 3374 106 4 v Fn(L) p Fl 632 3467 a(1) p Fo 672 3452 a(\)) p Ff(.) p 785 3452 a(In) p 912 3452 a(p) p 957 3452 a(articular,) p 1387 3452 a(the) p 1549 3452 a(element) p Fn 1910 3452 a(u) p Fk 1994 3452 a(2) p Fn 2088 3452 a(D) p Fo 2172 3452 a(\() p Fn(L) p Fg 2276 3416 a(\003) p Fl 2276 3477 a(1) p Fo 2315 3452 a(\)) p Ff 2388 3452 a(b) p 2428 3452 a(elongs) p 2722 3452 a(to) p Fn 2839 3452 a(D) p Fo 2923 3452 a(\() p 2961 3374 V Fn(L) p Fl 3027 3467 a(1) p Fo 3066 3452 a(\)) p Ff 3139 3452 a(if) p 3234 3452 a(and) p 3423 3452 a(only) p 3635 3452 a(if) p Fo 1135 3630 a(\010) p Fg 1205 3645 a(\000) p Fl(1) p Fo 1300 3630 a(\() p Fn(u) p Fo(\)) p 1459 3630 a(=) p 1563 3630 a(\011) p Fl 1639 3645 a(1) p Fo 1678 3630 a(\() p Fn(u) p Fo(\)) p 1837 3630 a(=) p 1941 3630 a(\010) p Fl 2011 3645 a(0) p Fo 2050 3630 a(\() p Fn(u) p Fo(\)) p 2209 3630 a(=) p 2313 3630 a(\011) p Fl 2389 3645 a(0) p Fo 2428 3630 a(\() p Fn(u) p Fo(\)) p 2587 3630 a(=) p 2691 3630 a(0) p Fn(:) p Fo 0 3807 a(Remark) p 367 3807 a(1.) p 487 3807 a(The) p 687 3807 a(four) p 891 3807 a(functions) p Fn 1311 3807 a(\036) p Fg 1369 3822 a(\000) p Fl(1) p Fo 1464 3807 a(,) p Fn 1523 3807 a( ) p Fl 1586 3822 a(1) p Fo 1626 3807 a(,) p Fn 1686 3807 a(\036) p Fl 1744 3822 a(0) p Fo 1783 3807 a(,) p Fn 1843 3807 a( ) p Fl 1906 3822 a(0) p Fo 1978 3807 a(are) p 2141 3807 a(eigenfunctions) p 2778 3807 a(of) p Fk 2890 3807 a(L) p Fl 2959 3822 a(1) p Fo 3031 3807 a(b) m(y) p 3166 3807 a(\(3.2\).) p 3437 3807 a(Th) m(us) p 3685 3807 a(these) 0 3927 y(functions) p 421 3927 a(b) s(elong) p 732 3927 a(to) p Fn 851 3927 a(D) p Fo 935 3927 a(\() p Fn(L) p Fg 1039 3891 a(\003) p Fl 1039 3952 a(1) p Fo 1079 3927 a(\)) p 1149 3927 a(b) m(y) p 1285 3927 a(\(i\)) p 1420 3927 a(of) p 1531 3927 a(this) p 1722 3927 a(prop) s(osition.) 0 4047 y(Remark) p 364 4047 a(2.) p 482 4047 a(The) p 679 4047 a(decomp) s(osition) p 1318 4047 a(\(3.4\)) p 1548 4047 a(is) p 1643 4047 a(a) p 1720 4047 a(paraphrase) p 2216 4047 a(of) p 2324 4047 a([Ex-St-Vy) p 2742 4047 a(,) p 2799 4047 a(page2158,) p 3246 4047 a(line8],) p 3531 4047 a(since) p 3767 4047 a(the) 0 4168 y(four) p 203 4168 a(functions) p 624 4168 a(ha) m(v) m(e) p 849 4168 a(the) p 1017 4168 a(asymptotics) p Fn 887 4345 a(\036) p Fg 945 4360 a(\000) p Fl(1) p Fk 1066 4345 a(\030) p Fn 1172 4345 a(r) p Fm 1219 4304 a(\013) p Fg(\000) p Fl(1) p Fn 1358 4345 a(e) p Fg 1403 4304 a(\000) p Fm(i\022) p Fn 1521 4345 a(;) p 1597 4345 a( ) p Fl 1660 4360 a(1) p Fk 1728 4345 a(\030) p Fn 1833 4345 a(r) p Fl 1880 4304 a(1) p Fg(\000) p Fm(\013) p Fn 2019 4345 a(e) p Fg 2064 4304 a(\000) p Fm(i\022) p Fn 2182 4345 a(;) p 2226 4345 a(\036) p Fl 2284 4360 a(0) p Fk 2351 4345 a(\030) p Fn 2456 4345 a(r) p Fm 2503 4304 a(\013) p Fn 2552 4345 a(;) p 2629 4345 a( ) p Fl 2692 4360 a(0) p Fk 2759 4345 a(\030) p Fn 2864 4345 a(r) p Fg 2911 4304 a(\000) p Fm(\013) p Fo 0 4522 a(as) p Fn 120 4522 a(r) p Fk 194 4522 a(!) p Fo 321 4522 a(0.) 146 4642 y(W) p 238 4642 a(e) p 313 4642 a(shall) p 540 4642 a(prepare) p 891 4642 a(a) p 971 4642 a(lemma) p 1284 4642 a(for) p 1432 4642 a(the) p 1599 4642 a(future) p 1888 4642 a(use.) p 2094 4642 a(In) p 2214 4642 a(the) p 2381 4642 a(sequel,) p 2698 4642 a(the) p 2865 4642 a(note) p Fk 3080 4642 a(jj) p Fn(u) p Fk(jj) p Fo 3278 4642 a(is) p 3375 4642 a(the) p Fn 3541 4642 a(L) p Fl 3607 4606 a(2) p Fo 3647 4642 a(-norm) p Fk 0 4763 a(jj) p Fn(u) p Fk(jj) p Fl 168 4727 a(2) p Fo 233 4763 a(=) p Fi 337 4692 a(R) p Fk 409 4763 a(j) p Fn(u) p Fk(j) p Fl 521 4727 a(2) p Fn 559 4763 a(dxdy) p Fo 768 4763 a(.) p 838 4763 a(W) p 930 4763 a(e) p 1005 4763 a(regard) p Fn 1309 4763 a(D) p Fo 1393 4763 a(\() p Fn(L) p Fg 1497 4727 a(\003) p Fl 1497 4787 a(1) p Fo 1536 4763 a(\)) p 1607 4763 a(as) p 1727 4763 a(a) p 1808 4763 a(Hilb) s(ert) p 2144 4763 a(space) p 2405 4763 a(equipp) s(ed) p 2822 4763 a(with) p 3044 4763 a(the) p 3212 4763 a(graph) p 3488 4763 a(norm) p Fk 1190 4940 a(jj) p Fn(u) p Fk(jj) p Fl 1358 4899 a(2) 1358 4964 y(1) p Fo 1423 4940 a(=) p Fk 1527 4940 a(jjL) p Fl 1652 4955 a(1) p Fn 1690 4940 a(u) p Fk(jj) p Fl 1802 4899 a(2) p Fo 1863 4940 a(+) p Fk 1961 4940 a(jj) p Fn(u) p Fk(jj) p Fl 2129 4899 a(2) p Fn 2166 4940 a(;) p 2242 4940 a(u) p Fk 2326 4940 a(2) p Fn 2420 4940 a(D) p Fo 2504 4940 a(\() p Fn(L) p Fg 2608 4899 a(\003) p Fl 2608 4964 a(1) p Fo 2647 4940 a(\)) p Fn(:) p Fo 1902 5214 a(16) p eop %%Page: 17 17 17 16 bop Fq 146 631 a(Lemma) p 543 631 a(3.3) p Ff 735 631 a(\(i\)) p 880 631 a(The) p 1079 631 a(fol) p 1189 631 a(lowing) p 1493 631 a(e) p 1533 631 a(qualities) p 1916 631 a(hold.) p Fk 841 891 a(jj) p Fn(\036) p Fm 955 850 a(\013) p Fg 955 916 a(\000) p Fl(1) p Fk 1049 891 a(jj) p Fl 1105 850 a(2) 1105 916 y(1) p Fo 1226 891 a(=) p Fn 1385 891 a(\031) p Fi 1461 795 a(n) p Fo 1516 891 a(\(2) p Fn(\013) p Fk 1688 891 a(\000) p Fo 1787 891 a(1\)) p Fl 1874 850 a(2) p Fn 1913 891 a(B) p Fl 1992 850 a(2) p Fo 2054 891 a(+) p 2152 891 a(1) p Fi 2201 795 a(o) 2273 770 y(\022) p Fo 2359 824 a(2) p 2344 868 80 4 v Fn 2344 959 a(B) p Fi 2433 770 a(\023) p Fm 2494 786 a(\013) p Fo 2560 891 a(\000\() p Fn(\013) p Fo 2722 891 a(\)) p Fn(;) p Fk 877 1147 a(jj) p Fn( ) p Fm 1000 1106 a(\013) p Fl 996 1172 a(1) p Fk 1049 1147 a(jj) p Fl 1105 1106 a(2) 1105 1172 y(1) p Fo 1226 1147 a(=) p Fn 1385 1147 a(\031) p Fo 1444 1147 a(\() p Fn(B) p Fl 1561 1106 a(2) p Fo 1623 1147 a(+) p 1721 1147 a(1\)) p Fi 1825 1026 a(\022) p Fo 1910 1080 a(2) p 1895 1124 V Fn 1895 1215 a(B) p Fi 1984 1026 a(\023) p Fl 2045 1042 a(2) p Fg(\000) p Fm(\013) p Fo 2202 1147 a(\000\(2) p Fk 2371 1147 a(\000) p Fn 2471 1147 a(\013) p Fo 2534 1147 a(\)) p Fn(;) p Fk 886 1403 a(jj) p Fn(\036) p Fm 1000 1362 a(\013) p Fl 1000 1428 a(0) p Fk 1049 1403 a(jj) p Fl 1105 1362 a(2) 1105 1428 y(1) p Fo 1226 1403 a(=) p Fn 1385 1403 a(\031) p Fi 1461 1307 a(n) p Fo 1516 1403 a(\(2) p Fn(\013) p Fo 1688 1403 a(+) p 1786 1403 a(1\)) p Fl 1873 1362 a(2) p Fn 1912 1403 a(B) p Fl 1991 1362 a(2) p Fo 2052 1403 a(+) p 2150 1403 a(1) p Fi 2199 1307 a(o) 2271 1282 y(\022) p Fo 2357 1336 a(2) p 2342 1380 V Fn 2342 1471 a(B) p Fi 2431 1282 a(\023) p Fl 2493 1298 a(1+) p Fm(\013) p Fo 2649 1403 a(\000\(1) p 2819 1403 a(+) p Fn 2917 1403 a(\013) p Fo 2980 1403 a(\)) p Fn(;) p Fk 877 1659 a(jj) p Fn( ) p Fm 1000 1618 a(\013) p Fl 996 1684 a(0) p Fk 1049 1659 a(jj) p Fl 1105 1618 a(2) 1105 1684 y(1) p Fo 1226 1659 a(=) p Fn 1385 1659 a(\031) p Fo 1444 1659 a(\() p Fn(B) p Fl 1561 1618 a(2) p Fo 1623 1659 a(+) p 1721 1659 a(1\)) p Fi 1825 1538 a(\022) p Fo 1910 1592 a(2) p 1895 1636 V Fn 1895 1727 a(B) p Fi 1984 1538 a(\023) p Fl 2045 1554 a(1) p Fg(\000) p Fm(\013) p Fo 2202 1659 a(\000\(1) p Fk 2371 1659 a(\000) p Fn 2471 1659 a(\013) p Fo 2534 1659 a(\)) p Fn(;) p Ff 0 1914 a(wher) p 196 1914 a(e) p Fo 275 1914 a(\000\() p Fn(z) p Fo 423 1914 a(\)) p 489 1914 a(=) p Fi 593 1843 a(R) p Fg 648 1869 a(1) p Fl 632 1939 a(0) p Fn 740 1914 a(e) p Fg 785 1878 a(\000) p Fm(t) p Fn 869 1914 a(t) p Fm 904 1878 a(z) p Fg 940 1878 a(\000) p Fl(1) p Fn 1035 1914 a(dt) p Ff 1155 1914 a(is) p 1260 1914 a(the) p 1422 1914 a(Gamma) p 1791 1914 a(function.) 146 2034 y(\(ii\)) p 321 2034 a(F) p 378 2034 a(or) p Fn 503 2034 a(u;) p 603 2034 a(v) p Fk 680 2034 a(2) p Fn 774 2034 a(D) p Fo 858 2034 a(\(\() p Fn(L) p Fm 1000 1998 a(\013) p Fl 1000 2059 a(1) p Fo 1050 2034 a(\)) p Fg 1088 1998 a(\003) p Fo 1127 2034 a(\)) p Ff(,) p 1230 2034 a(de\014ne) p Fo 1345 2247 a([) p Fn(u;) p 1472 2247 a(v) p Fo 1523 2247 a(]) p Fl 1550 2262 a(1) p Fo 1616 2247 a(=) p 1720 2247 a(\() p Fk(L) p Fm 1827 2206 a(\013) p Fl 1827 2272 a(1) p Fn 1876 2247 a(u;) p 1976 2247 a(v) p Fo 2027 2247 a(\)) p Fk 2086 2247 a(\000) p Fo 2186 2247 a(\() p Fn(u;) p Fk 2324 2247 a(L) p Fm 2393 2206 a(\013) p Fl 2393 2272 a(1) p Fn 2441 2247 a(v) p Fo 2492 2247 a(\)) p Fn(:) p Ff 0 2461 a(Then,) p 284 2461 a(the) p 446 2461 a(fol) p 556 2461 a(lowing) p 859 2461 a(e) p 899 2461 a(qualities) p 1283 2461 a(hold.) p Fo 980 2674 a([) p Fn(\036) p Fm 1065 2633 a(\013) p Fg 1065 2699 a(\000) p Fl(1) p Fn 1159 2674 a(;) p 1203 2674 a( ) p Fm 1270 2633 a(\013) p Fl 1266 2699 a(1) p Fo 1320 2674 a(]) p Fl 1347 2689 a(1) p Fo 1414 2674 a(=) p 1517 2674 a(4) p Fn(\031) p Fo 1625 2674 a(\() p Fn(\013) p Fk 1748 2674 a(\000) p Fo 1847 2674 a(1\)) p Fn(;) p Fo 1978 2674 a([) p Fn(\036) p Fm 2063 2633 a(\013) p Fl 2063 2699 a(0) p Fn 2112 2674 a(;) p 2156 2674 a( ) p Fm 2223 2633 a(\013) p Fl 2219 2699 a(0) p Fo 2272 2674 a(]) p Fl 2299 2689 a(1) p Fo 2367 2674 a(=) p 2470 2674 a(4) p Fn(\031) t(\013) q(;) p Fo 980 2819 a([) p Fn(\036) p Fm 1065 2778 a(\013) p Fg 1065 2844 a(\000) p Fl(1) p Fn 1159 2819 a(;) p 1203 2819 a(\036) p Fm 1261 2778 a(\013) p Fl 1261 2844 a(0) p Fo 1310 2819 a(]) p Fl 1337 2834 a(1) p Fo 1405 2819 a(=) p 1508 2819 a([) p Fn(\036) p Fm 1593 2778 a(\013) p Fg 1593 2844 a(\000) p Fl(1) p Fn 1687 2819 a(;) p 1731 2819 a( ) p Fm 1798 2778 a(\013) p Fl 1794 2844 a(0) p Fo 1847 2819 a(]) p Fl 1874 2834 a(1) p Fo 1942 2819 a(=) p 2045 2819 a([) p Fn( ) p Fm 2139 2778 a(\013) p Fl 2135 2844 a(1) p Fn 2189 2819 a(;) p 2233 2819 a(\036) p Fm 2291 2778 a(\013) p Fl 2291 2844 a(0) p Fo 2340 2819 a(]) p Fl 2367 2834 a(1) p Fo 2434 2819 a(=) p 2537 2819 a([) p Fn( ) p Fm 2631 2778 a(\013) p Fl 2627 2844 a(1) p Fn 2681 2819 a(;) p 2725 2819 a( ) p Fm 2792 2778 a(\013) p Fl 2788 2844 a(0) p Fo 2841 2819 a(]) p Fl 2868 2834 a(1) p Fo 2935 2819 a(=) p 3039 2819 a(0) 980 2964 y([) p Fn(\036) p Fm 1065 2923 a(\013) 1065 2989 y(l) p Fn 1115 2964 a(;) p 1159 2964 a(\036) p Fm 1217 2923 a(\013) 1217 2989 y(l) p Fo 1266 2964 a(]) p Fl 1293 2979 a(1) p Fo 1360 2964 a(=) p 1463 2964 a([) p Fn( ) p Fm 1557 2923 a(\013) 1553 2989 y(n) p Fn 1607 2964 a(;) p 1651 2964 a( ) p Fm 1718 2923 a(\013) 1714 2989 y(n) p Fo 1767 2964 a(]) p Fl 1794 2979 a(1) p Fo 1861 2964 a(=) p 1965 2964 a(0) p 2118 2964 a(for) p Fn 2270 2964 a(l) p Fo 2328 2964 a(=) p Fk 2432 2964 a(\000) p Fo(1) p Fn(;) p Fo 2602 2964 a(0) p Fn(;) p 2729 2964 a(n) p Fo 2815 2964 a(=) p 2919 2964 a(0) p Fn(;) p Fo 3012 2964 a(1) p Fn(:) p Ff 146 3178 a(\(iii\)) p 359 3178 a(The) p 567 3178 a(op) p 662 3178 a(er) p 743 3178 a(ators) p Fo 999 3178 a(\010) p Fm 1069 3142 a(\013) p Fg 1069 3202 a(\000) p Fl(1) p Ff 1163 3178 a(,) p Fo 1239 3178 a(\011) p Fm 1315 3142 a(\013) p Fl 1315 3202 a(1) p Ff 1364 3178 a(,) p Fo 1439 3178 a(\010) p Fm 1509 3142 a(\013) p Fl 1509 3202 a(0) p Ff 1559 3178 a(,) p Fo 1634 3178 a(\011) p Fm 1710 3142 a(\013) p Fl 1710 3202 a(0) p Ff 1803 3178 a(ar) p 1889 3178 a(e) p 1977 3178 a(b) p 2017 3178 a(ounde) p 2264 3178 a(d) p 2357 3178 a(line) p 2507 3178 a(ar) p 2640 3178 a(functionals) p 3147 3178 a(on) p Fn 3295 3178 a(D) p Fo 3379 3178 a(\() p Fn(L) p Fm 3483 3142 a(\013) p Fl 3483 3202 a(1) p Fo 3532 3178 a(\)) p Fg 3570 3142 a(\003) p Ff 3653 3178 a(whose) 0 3298 y(op) p 95 3298 a(er) p 176 3298 a(ator) p 384 3298 a(norm) p 644 3298 a(ar) p 730 3298 a(e) p 809 3298 a(estimate) p 1158 3298 a(d) p 1243 3298 a(by) p Fk 773 3550 a(jj) p Fo(\010) p Fm 899 3514 a(\013) p Fg 899 3575 a(\000) p Fl(1) p Fk 993 3550 a(jj) p 1076 3550 a(\024) p Fo 1375 3483 a(1) p 1191 3527 417 4 v 1191 3619 a(4) p Fn(\031) p Fo 1299 3619 a(\(1) p Fk 1408 3619 a(\000) p Fn 1507 3619 a(\013) p Fo 1570 3619 a(\)) p Fk 1618 3550 a(jj) p Fn( ) p Fm 1741 3509 a(\013) p Fl 1737 3575 a(1) p Fk 1790 3550 a(jj) p Fl 1846 3565 a(1) p Fn 1884 3550 a(;) p Fk 1994 3550 a(jj) p Fo(\011) p Fm 2126 3514 a(\013) p Fl 2126 3575 a(1) p Fk 2175 3550 a(jj) p 2258 3550 a(\024) p Fo 2557 3483 a(1) p 2373 3527 V 2373 3619 a(4) p Fn(\031) p Fo 2481 3619 a(\(1) p Fk 2590 3619 a(\000) p Fn 2689 3619 a(\013) p Fo 2752 3619 a(\)) p Fk 2800 3550 a(jj) p Fn(\036) p Fm 2914 3509 a(\013) p Fg 2914 3575 a(\000) p Fl(1) p Fk 3007 3550 a(jj) p Fl 3063 3565 a(1) p Fn 3102 3550 a(;) p Fk 773 3775 a(jj) p Fo(\010) p Fm 899 3739 a(\013) p Fl 899 3800 a(0) p Fk 948 3775 a(jj) p 1031 3775 a(\024) p Fo 1207 3708 a(1) p 1146 3752 171 4 v 1146 3843 a(4) p Fn(\031) t(\013) p Fk 1326 3775 a(jj) p Fn( ) p Fm 1449 3734 a(\013) p Fl 1445 3800 a(0) p Fk 1498 3775 a(jj) p Fl 1554 3790 a(1) p Fn 1593 3775 a(;) p Fk 1994 3775 a(jj) p Fo(\011) p Fm 2126 3739 a(\013) p Fl 2126 3800 a(0) p Fk 2175 3775 a(jj) p 2258 3775 a(\024) p Fo 2434 3708 a(1) p 2373 3752 V 2373 3843 a(4) p Fn(\031) t(\013) p Fk 2554 3775 a(jj) p Fn(\036) p Fm 2668 3734 a(\013) p Fl 2668 3800 a(0) p Fk 2716 3775 a(jj) p Fl 2772 3790 a(1) p Fn 2811 3775 a(:) p Ff 146 4027 a(Pr) p 248 4027 a(o) p 293 4027 a(of.) p Fo 434 4027 a(\(i\)) p 568 4027 a(\(ii\)) p 728 4027 a(One) p 933 4027 a(can) p 1110 4027 a(pro) m(v) m(e) p 1371 4027 a(these) p 1619 4027 a(equalities) p 2049 4027 a(b) m(y) p 2182 4027 a(a) p 2262 4027 a(short) p 2510 4027 a(calculation) p 3002 4027 a(using) p 3255 4027 a(\(3.2\)) p 3486 4027 a(and) p 3674 4027 a(\(3.3\).) 146 4147 y(\(iii\)) p 336 4147 a(Notice) p 642 4147 a(that) 1281 4267 y([) p Fn(u;) p 1408 4267 a(v) p Fo 1459 4267 a(]) p Fl 1486 4282 a(1) p Fo 1552 4267 a(=) p 1656 4267 a(\() p Fn(L) p Fg 1760 4226 a(\003) p Fl 1760 4292 a(1) p Fn 1800 4267 a(u;) p 1900 4267 a(v) p Fo 1951 4267 a(\)) p Fk 2010 4267 a(\000) p Fo 2109 4267 a(\() p Fn(u;) p 2247 4189 106 4 v 2247 4267 a(L) p Fl 2313 4282 a(1) p Fn 2352 4267 a(v) p Fo 2403 4267 a(\)) p 2469 4267 a(=) p 2572 4267 a(0) 0 4439 y(for) p 149 4439 a(an) m(y) p Fn 333 4439 a(u) p Fk 416 4439 a(2) p Fn 510 4439 a(D) p Fo 594 4439 a(\() p Fn(L) p Fg 698 4403 a(\003) p Fl 698 4463 a(1) p Fo 737 4439 a(\)) p 807 4439 a(and) p Fn 997 4439 a(v) p Fk 1075 4439 a(2) p Fn 1169 4439 a(D) p Fo 1253 4439 a(\() p 1291 4361 V Fn(L) p Fl 1357 4454 a(1) p Fo 1397 4439 a(\).) p 1505 4439 a(F) p 1561 4439 a(rom) p 1761 4439 a(this) p 1950 4439 a(equalit) m(y) p 2281 4439 a(,) p 2340 4439 a(\(ii\)) p 2503 4439 a(of) p 2613 4439 a(this) p 2803 4439 a(lemma) p 3117 4439 a(and) p 3307 4439 a(\(3.4\),) p 3566 4439 a(w) m(e) p 3710 4439 a(ha) m(v) m(e) 711 4696 y(\010) p Fm 781 4660 a(\013) p Fg 781 4721 a(\000) p Fl(1) p Fo 876 4696 a(\() p Fn(u) p Fo(\)) p 1035 4696 a(=) 1332 4629 y(1) p 1148 4673 417 4 v 1148 4765 a(4) p Fn(\031) p Fo 1256 4765 a(\(1) p Fk 1365 4765 a(\000) p Fn 1464 4765 a(\013) p Fo 1527 4765 a(\)) 1575 4696 y([) p Fn( ) p Fm 1669 4655 a(\013) p Fl 1665 4721 a(1) p Fn 1719 4696 a(;) p 1763 4696 a(u) p Fo(]) p Fl 1846 4711 a(1) p Fn 1884 4696 a(;) p Fo 1994 4696 a(\011) p Fm 2070 4660 a(\013) p Fl 2070 4721 a(1) p Fo 2120 4696 a(\() p Fn(u) p Fo(\)) p 2279 4696 a(=) 2576 4629 y(1) p 2392 4673 V 2392 4765 a(4) p Fn(\031) p Fo 2500 4765 a(\() p Fn(\013) p Fk 2623 4765 a(\000) p Fo 2722 4765 a(1\)) 2819 4696 y([) p Fn(\036) p Fm 2904 4655 a(\013) p Fg 2904 4721 a(\000) p Fl(1) p Fn 2998 4696 a(;) p 3042 4696 a(u) p Fo(]) p Fl 3125 4711 a(1) p Fn 3164 4696 a(;) p Fo 711 4921 a(\010) p Fm 781 4885 a(\013) p Fl 781 4946 a(0) p Fo 831 4921 a(\() p Fn(u) p Fo(\)) p 990 4921 a(=) 1203 4854 y(1) p 1103 4898 248 4 v Fk 1103 4990 a(\000) p Fo(4) p Fn(\031) t(\013) p Fo 1361 4921 a([) p Fn( ) p Fm 1455 4880 a(\013) p Fl 1451 4946 a(0) p Fn 1505 4921 a(;) p 1549 4921 a(u) p Fo(]) p Fl 1632 4936 a(1) p Fn 1671 4921 a(;) p Fo 1994 4921 a(\011) p Fm 2070 4885 a(\013) p Fl 2070 4946 a(0) p Fo 2120 4921 a(\() p Fn(u) p Fo(\)) p 2279 4921 a(=) 2453 4854 y(1) p 2392 4898 171 4 v 2392 4990 a(4) p Fn(\031) t(\013) p Fo 2573 4921 a([) p Fn(\036) p Fm 2658 4880 a(\013) p Fl 2658 4946 a(0) p Fn 2707 4921 a(;) p 2751 4921 a(u) p Fo(]) p Fl 2834 4936 a(1) p Fn 2873 4921 a(:) p Fo 3701 4806 a(\(3.5\)) 1902 5214 y(17) p eop %%Page: 18 18 18 17 bop Fo 0 631 a(Moreo) m(v) m(er,) p 455 631 a(w) m(e) p 599 631 a(ha) m(v) m(e) p 824 631 a(b) m(y) p 959 631 a(the) p 1127 631 a(Sc) m(h) m(w) m(arz) p 1504 631 a(inequalit) m(y) p Fk 937 839 a(j) p Fo([) p Fn(u;) p 1092 839 a(v) p Fo 1143 839 a(]) p Fl 1170 854 a(1) p Fk 1208 839 a(j) p 1264 839 a(\024) p 1369 839 a(jjL) p Fl 1494 854 a(1) p Fn 1532 839 a(u) p Fk(jj) p 1676 839 a(jj) p Fn(v) p Fk 1783 839 a(jj) p Fo 1859 839 a(+) p Fk 1957 839 a(jj) p Fn(u) p Fk(jj) p 2156 839 a(jjL) p Fl 2281 854 a(1) p Fn 2320 839 a(v) p Fk 2371 839 a(jj) p 2453 839 a(\024) p 2558 839 a(jj) p Fn(u) p Fk(jj) p Fl 2726 854 a(1) p Fk 2764 839 a(jj) p Fn(v) p Fk 2871 839 a(jj) p Fl 2927 854 a(1) p Fo 3701 839 a(\(3.6\)) 0 1048 y(for) p Fn 149 1048 a(u;) p 249 1048 a(v) p Fk 327 1048 a(2) p Fn 421 1048 a(D) p Fo 505 1048 a(\() p Fn(L) p Fl 609 1063 a(1) p Fo 648 1048 a(\)) p Fg 686 1012 a(\003) p Fo 726 1048 a(.) p 796 1048 a(By) p 949 1048 a(\(3.5\)) p 1182 1048 a(and) p 1372 1048 a(\(3.6\),) p 1632 1048 a(the) p 1800 1048 a(conclusion) p 2272 1048 a(holds.) p 2565 980 64 4 v 2565 1073 4 93 v 2625 1073 V 2565 1076 64 4 v Fh 0 1379 a(4) p 242 1379 a(De\014ciency) p 1020 1379 a(indices) p 1551 1379 a(of) p Fc 1735 1379 a(L) p Fn 1831 1401 a(N) p Fo 0 1598 a(In) p 129 1598 a(this) p 327 1598 a(section,) p 689 1598 a(w) m(e) p 839 1598 a(shall) p 1075 1598 a(determine) p 1537 1598 a(the) p 1713 1598 a(de\014ciency) p 2167 1598 a(indices) p 2494 1598 a(of) p Fn 2613 1598 a(L) p Fm 2679 1613 a(N) p Fo 2746 1598 a(.) p 2839 1598 a(W) p 2931 1598 a(e) p 3014 1598 a(shall) p 3249 1598 a(in) m(tro) s(duce) p 3690 1598 a(some) 0 1718 y(di\013eren) m(tial) p 488 1718 a(op) s(erators) p 919 1718 a(con) m(v) m(enien) m(t) p 1401 1718 a(for) p 1551 1718 a(later) p 1778 1718 a(calculus.) p Fq 146 1912 a(Lemma) p 543 1912 a(4.1) p Ff 735 1912 a(De\014ne) p 1033 1912 a(di\013er) p 1254 1912 a(ential) p 1515 1912 a(op) p 1610 1912 a(er) p 1691 1912 a(ators) p Fo 1929 1912 a(\005) p Fm 2002 1927 a(N) s(;x) p Fn 2124 1912 a(;) p Fo 2168 1912 a(\005) p Fm 2241 1927 a(N) s(;y) p Fn 2361 1912 a(;) p Fk 2405 1912 a(A) p Fm 2485 1927 a(N) p Fn 2552 1912 a(;) p Fk 2596 1912 a(A) p Fg 2676 1865 a(y) p Fm 2676 1936 a(N) p Ff 2767 1912 a(fr) p 2833 1912 a(om) p Fk 2988 1912 a(D) p Fg 3068 1876 a(0) p Fo 3091 1912 a(\() p Fj(R) p Fl 3216 1870 a(2) p Fk 3255 1912 a(n) p Fn(S) p Fm 3365 1927 a(N) p Fo 3432 1912 a(\)) p Ff 3495 1912 a(to) p Fk 3602 1912 a(D) p Fg 3682 1876 a(0) p Fo 3705 1912 a(\() p Fj(R) p Fl 3830 1870 a(2) p Fk 3870 1912 a(n) p Fn 0 2032 a(S) p Fm 60 2047 a(N) p Fo 127 2032 a(\)) p Ff 200 2032 a(by) p Fo 1082 2285 a(\005) p Fm 1155 2300 a(N) s(;x) p Fo 1361 2285 a(=) 1530 2218 y(1) p 1530 2262 49 4 v Fn 1538 2354 a(i) 1589 2285 y(@) p Fm 1640 2300 a(x) p Fo 1706 2285 a(+) p Fn 1804 2285 a(a) p Fm 1855 2300 a(N) s(;x) p Fn 1978 2285 a(;) p Fo 1085 2518 a(\005) p Fm 1158 2533 a(N) s(;y) p Fo 1361 2518 a(=) 1530 2451 y(1) p 1530 2495 V Fn 1538 2587 a(i) 1589 2518 y(@) p Fm 1640 2533 a(y) p Fo 1704 2518 a(+) p Fn 1802 2518 a(a) p Fm 1853 2533 a(N) s(;y) p Fn 1973 2518 a(;) p Fk 1131 2695 a(A) p Fm 1211 2710 a(N) p Fo 1361 2695 a(=) p Fn 1520 2695 a(i) p Fo(\005) p Fm 1626 2710 a(N) s(;x) p Fo 1771 2695 a(+) p 1869 2695 a(\005) p Fm 1942 2710 a(N) s(;y) p Fo 2090 2695 a(=) p 2194 2695 a(2) p Fn(@) p Fm 2294 2710 a(z) p Fo 2356 2695 a(+) p Fn 2454 2695 a(\036) p Fo(\() p Fn(z) p Fo 2599 2695 a(\)) p Fn(;) p Fk 1131 2849 a(A) p Fg 1211 2802 a(y) p Fm 1211 2873 a(N) p Fo 1361 2849 a(=) p Fk 1520 2849 a(\000) p Fn(i) p Fo(\005) p Fm 1703 2864 a(N) s(;x) p Fo 1849 2849 a(+) p 1947 2849 a(\005) p Fm 2020 2864 a(N) s(;y) p Fo 2168 2849 a(=) p Fk 2271 2849 a(\000) p Fo(2) p Fn(@) p Fl 2452 2864 a(\026) p Fm 2448 2864 a(z) p Fo 2511 2849 a(+) p 2609 2764 184 4 v Fn 2609 2849 a(\036) p Fo(\() p Fn(z) p Fo 2754 2849 a(\)) p Fn(;) p Ff 0 3058 a(wher) p 196 3058 a(e) p Fn 274 3058 a(\036) p Fo(\() p Fn(z) p Fo 419 3058 a(\)) p 485 3058 a(=) p Fn 589 3058 a(ia) p Fm 673 3073 a(N) s(;x) p Fo 796 3058 a(\() p Fn(z) p Fo 883 3058 a(\)) p 941 3058 a(+) p Fn 1036 3058 a(a) p Fm 1087 3073 a(N) s(;y) p Fo 1207 3058 a(\() p Fn(z) p Fo 1294 3058 a(\)) p Ff(.) p 1407 3058 a(L) p 1463 3058 a(et) p Fn 1574 3058 a(t) p Fg 1609 3073 a(\000) p Fm(z) p Fe 1697 3083 a(j) p Ff 1768 3058 a(b) p 1808 3058 a(e) p 1886 3058 a(the) p 2047 3058 a(magnetic) p 2461 3058 a(tr) p 2529 3058 a(anslation) p 2949 3058 a(fr) p 3015 3058 a(om) p Fk 3179 3058 a(fj) p Fn(z) p Fk 3325 3058 a(\000) p Fn 3422 3058 a(z) p Fm 3467 3073 a(j) p Fk 3504 3058 a(j) p Fn 3559 3058 a(<) p Fm 3673 3018 a(R) p 3673 3034 54 4 v Fl 3682 3092 a(2) p Fk 3736 3058 a(g) p Ff 3820 3058 a(to) p Fk 0 3178 a(fj) p Fn(z) p Fk 127 3178 a(j) p Fn 183 3178 a(<) p Fm 296 3139 a(R) p 296 3155 V Fl 305 3212 a(2) p Fk 360 3178 a(g) p Ff 444 3178 a(intertwining) p Fk 994 3178 a(L) p Fm 1063 3193 a(N) p Ff 1164 3178 a(with) p Fk 1376 3178 a(L) p Fl 1445 3193 a(1) p Ff 1484 3178 a(.) p 1559 3178 a(Then) p 1813 3178 a(the) p 1975 3178 a(fol) p 2085 3178 a(lowing) p 2388 3178 a(e) p 2428 3178 a(qualities) p 2812 3178 a(holds:) p Fk 1137 3402 a(A) p Fg 1217 3355 a(y) p Fm 1217 3426 a(N) p Fk 1284 3402 a(A) p Fm 1364 3417 a(N) p Fo 1459 3402 a(=) p Fk 1562 3402 a(L) p Fm 1631 3417 a(N) p Fk 1720 3402 a(\000) p Fn 1820 3402 a(B) p 1899 3402 a(;) p Fk 1978 3402 a(A) p Fm 2058 3417 a(N) p Fk 2124 3402 a(A) p Fg 2204 3355 a(y) p Fm 2204 3426 a(N) p Fo 2299 3402 a(=) p Fk 2402 3402 a(L) p Fm 2471 3417 a(N) p Fo 2560 3402 a(+) p Fn 2658 3402 a(B) p 2737 3402 a(;) p Fo 3701 3402 a(\(4.1\)) p Ff 0 3611 a(as) p 125 3611 a(op) p 220 3611 a(er) p 301 3611 a(ators) p 548 3611 a(fr) p 614 3611 a(om) p Fk 778 3611 a(D) p Fg 858 3574 a(0) p Fo 881 3611 a(\() p Fj(R) p Fl 1007 3569 a(2) p Fk 1068 3611 a(n) p Fn 1140 3611 a(S) p Fm 1200 3626 a(N) p Fo 1267 3611 a(\)) p Ff 1340 3611 a(to) p Fk 1457 3611 a(D) p Fg 1537 3574 a(0) p Fo 1560 3611 a(\() p Fj(R) p Fl 1685 3569 a(2) p Fk 1747 3611 a(n) p Fn 1819 3611 a(S) p Fm 1879 3626 a(N) p Fo 1946 3611 a(\)) p Ff(.) p Fo 1494 3819 a(\() p Fk(A) p Fm 1612 3834 a(N) p Fn 1679 3819 a(\036;) p 1781 3819 a( ) p Fo 1848 3819 a(\)) p 1913 3819 a(=) p 2017 3819 a(\() p Fn(\036;) p Fk 2157 3819 a(A) p Fg 2237 3772 a(y) p Fm 2237 3843 a(N) p Fn 2303 3819 a( ) p Fo 2370 3819 a(\)) p 3701 3819 a(\(4.2\)) p Ff 0 4028 a(for) p 156 4028 a(every) p Fn 413 4028 a(\036;) p 515 4028 a( ) p Fk 610 4028 a(2) p Fn 704 4028 a(C) p Fg 781 3991 a(1) p Fl 774 4052 a(0) p Fo 855 4028 a(\() p Fj(R) p Fl 981 3986 a(2) p Fk 1042 4028 a(n) p Fn 1114 4028 a(S) p Fm 1174 4043 a(N) p Fo 1241 4028 a(\)) p Ff(.) p Fk 1127 4249 a(A) p Fm 1207 4195 a(\013) p Fe 1252 4205 a(j) p Fl 1207 4271 a(1) p Fn 1289 4249 a(t) p Fg 1324 4264 a(\000) p Fm(z) p Fe 1412 4274 a(j) p Fo 1476 4249 a(=) p Fn 1580 4249 a(t) p Fg 1615 4264 a(\000) p Fm(z) p Fe 1703 4274 a(j) p Fk 1739 4249 a(A) p Fm 1819 4264 a(N) p Fn 1886 4249 a(;) p Fk 1965 4249 a(A) p Fg 2045 4195 a(y) p Fm 2076 4195 a(;\013) p Fe 2141 4205 a(j) p Fl 2045 4271 a(1) p Fn 2177 4249 a(t) p Fg 2212 4264 a(\000) p Fm(z) p Fe 2300 4274 a(j) p Fo 2365 4249 a(=) p Fn 2468 4249 a(t) p Fg 2503 4264 a(\000) p Fm(z) p Fe 2591 4274 a(j) p Fk 2628 4249 a(A) p Fg 2708 4202 a(y) p Fm 2708 4274 a(N) p Fo 3701 4249 a(\(4.3\)) p Ff 0 4458 a(as) p 125 4458 a(op) p 220 4458 a(er) p 301 4458 a(ators) p 548 4458 a(fr) p 614 4458 a(om) p Fk 778 4458 a(D) p Fg 858 4421 a(0) p Fo 881 4458 a(\() p Fk(fj) p Fn(z) p Fk 1069 4458 a(\000) p Fn 1168 4458 a(z) p Fm 1213 4473 a(j) p Fk 1250 4458 a(j) p Fn 1305 4458 a(<) p Fm 1419 4418 a(R) p 1419 4434 V Fl 1428 4492 a(2) p Fk 1482 4458 a(g) p Fo(\)) p Ff 1605 4458 a(to) p Fk 1722 4458 a(D) p Fg 1802 4421 a(0) p Fo 1825 4458 a(\() p Fk(fj) p Fn(z) p Fk 1990 4458 a(j) p Fn 2046 4458 a(<) p Fm 2159 4418 a(R) p 2159 4434 V Fl 2168 4492 a(2) p Fk 2223 4458 a(g) p Fo(\)) p Ff(.) 146 4673 y(Pr) p 248 4673 a(o) p 293 4673 a(of.) p Fo 435 4673 a(One) p 641 4673 a(can) p 820 4673 a(pro) m(v) m(e) p 1083 4673 a(\(4.1\)) p 1316 4673 a(b) m(y) p 1452 4673 a(a) p 1533 4673 a(short) p 1783 4673 a(computation) p 2352 4673 a(using) p 2607 4673 a(the) p 2775 4673 a(form) m(ula) 1153 4930 y([\005) p Fm 1253 4945 a(N) s(;x) p Fn 1375 4930 a(;) p Fo 1419 4930 a(\005) p Fm 1492 4945 a(N) s(;y) p Fo 1612 4930 a(]) p 1667 4930 a(=) p Fn 1781 4862 a(B) p 1781 4906 80 4 v 1804 4998 a(i) 1870 4930 y(;) p Fk 1946 4930 a(L) p Fm 2015 4945 a(N) p Fo 2110 4930 a(=) p 2213 4930 a(\005) p Fl 2286 4888 a(2) p Fm 2286 4954 a(N) s(;x) p Fo 2431 4930 a(+) p 2529 4930 a(\005) p Fl 2602 4888 a(2) p Fm 2602 4954 a(N) s(;y) p Fn 2722 4930 a(:) p Fo 1902 5214 a(18) p eop %%Page: 19 19 19 18 bop Fo 0 631 a(The) p 192 631 a(equalit) m(y) p 554 631 a(\(4.2\)) p 779 631 a(follo) m(ws) p 1090 631 a(from) p 1312 631 a(an) p 1439 631 a(in) m(tegration) p 1926 631 a(b) m(y) p 2053 631 a(parts.) p 2338 631 a(Notice) p 2635 631 a(that) p 2838 631 a(the) p 2997 631 a(in) m(tert) m(wining) p 3536 631 a(prop) s(ert) m(y) p Fi 1177 774 a(\022) p Fo 1248 828 a(1) p 1248 872 49 4 v Fn 1256 964 a(i) p Fk 1307 895 a(r) p Fo 1412 895 a(+) p Fj 1510 895 a(a) p Fm 1573 841 a(\013) p Fe 1618 851 a(j) p Fl 1573 917 a(1) p Fi 1655 774 a(\023) p Fn 1733 895 a(t) p Fg 1768 910 a(\000) p Fm(z) p Fe 1856 920 a(j) p Fo 1920 895 a(=) p Fn 2024 895 a(t) p Fg 2059 910 a(\000) p Fm(z) p Fe 2147 920 a(j) p Fi 2200 774 a(\022) p Fo 2271 828 a(1) p 2271 872 V Fn 2279 964 a(i) p Fk 2330 895 a(r) p Fo 2435 895 a(+) p Fj 2533 895 a(a) p Fm 2596 910 a(N) p Fi 2663 774 a(\023) p Fo 0 1147 a(implies) 1099 1268 y(\005) p Fm 1172 1226 a(\013) p Fe 1217 1236 a(j) p Fm 1172 1292 a(x) p Fn 1254 1268 a(t) p Fg 1289 1283 a(\000) p Fm(z) p Fe 1377 1293 a(j) p Fo 1442 1268 a(=) p Fn 1545 1268 a(t) p Fg 1580 1283 a(\000) p Fm(z) p Fe 1668 1293 a(j) p Fo 1705 1268 a(\005) p Fm 1778 1283 a(N) s(;x) p Fn 1901 1268 a(;) p Fo 1977 1268 a(\005) p Fm 2050 1226 a(\013) p Fe 2095 1236 a(j) p Fm 2050 1292 a(y) p Fn 2132 1268 a(t) p Fg 2167 1283 a(\000) p Fm(z) p Fe 2255 1293 a(j) p Fo 2319 1268 a(=) p Fn 2423 1268 a(t) p Fg 2458 1283 a(\000) p Fm(z) p Fe 2546 1293 a(j) p Fo 2582 1268 a(\005) p Fm 2655 1283 a(N) s(;y) p Fn 2776 1268 a(:) p Fo 0 1442 a(The) p 201 1442 a(equalit) m(y) p 572 1442 a(\(4.3\)) p 805 1442 a(follo) m(ws) p 1125 1442 a(from) p 1355 1442 a(these) p 1605 1442 a(equalities.) p 2075 1374 64 4 v 2075 1467 4 93 v 2135 1467 V 2075 1470 64 4 v 146 1670 a(In) p 268 1670 a(the) p 436 1670 a(sequel,) p 754 1670 a(w) m(e) p 897 1670 a(denote) p Fn 1247 1959 a(U) p Fo 1323 1959 a(\() p Fn(r) p Fo 1408 1959 a(\)) p 1474 1959 a(=) p Fm 1607 1851 a(N) p Fi 1592 1876 a([) p Fm 1577 2058 a(j) p Fl 1610 2058 a(=1) p Fk 1700 1959 a(f) p Fn(z) p Fk 1827 1959 a(2) p Fj 1921 1959 a(R) p Fl 2008 1917 a(2) p Fo 2048 1959 a(;) p Fk 2092 1959 a(j) p Fn(z) p Fk 2191 1959 a(\000) p Fn 2291 1959 a(z) p Fm 2336 1974 a(j) p Fk 2373 1959 a(j) p Fn 2428 1959 a(<) p 2532 1959 a(r) p Fk 2579 1959 a(g) p Fn(:) p Fq 146 2257 a(Prop) s(osition) p 752 2257 a(4.2) p Ff 944 2257 a(L) p 1000 2257 a(et) p Fn 1112 2257 a(N) p Fo 1228 2257 a(=) p 1332 2257 a(1) p Fn(;) p Fo 1425 2257 a(2) p Fn(;) p 1518 2257 a(:) p 1562 2257 a(:) p 1606 2257 a(:) p 1648 2257 a(;) p Ff 1710 2257 a(or) p Fn 1836 2257 a(N) p Fo 1952 2257 a(=) p Fk 2055 2257 a(1) p Ff(.) p 2230 2257 a(Then) p 2484 2257 a(the) p 2646 2257 a(fol) p 2756 2257 a(lowing) p 3059 2257 a(holds:) 146 2377 y(\(i\)) p 291 2377 a(The) p 490 2377 a(op) p 585 2377 a(er) p 666 2377 a(ator) p 874 2377 a(domain) p 1222 2377 a(of) p Fn 1337 2377 a(D) p Fo 1421 2377 a(\() p Fn(L) p Fg 1525 2341 a(\003) p Fm 1525 2402 a(N) p Fo 1592 2377 a(\)) p Ff 1665 2377 a(is) p 1770 2377 a(given) p 2024 2377 a(by) p Fn 729 2597 a(D) p Fo 813 2597 a(\() p Fn(L) p Fg 917 2556 a(\003) p Fm 917 2622 a(N) p Fo 985 2597 a(\)) p 1050 2597 a(=) p Fk 1154 2597 a(f) p Fn(u) p Fk 1287 2597 a(2) p Fn 1381 2597 a(L) p Fl 1447 2556 a(2) p Fo 1487 2597 a(\() p Fj(R) p Fl 1612 2555 a(2) p Fo 1651 2597 a(\)) p Fk 1711 2597 a(\\) p Fn 1800 2597 a(H) p Fl 1889 2556 a(2) p Fm 1881 2622 a(l) q(oc) p Fo 1972 2597 a(\() p Fj(R) p Fl 2097 2555 a(2) p Fk 2158 2597 a(n) p Fn 2230 2597 a(S) p Fm 2290 2612 a(N) p Fo 2358 2597 a(\);) p Fk 2474 2597 a(L) p Fm 2543 2612 a(N) p Fn 2610 2597 a(u) p Fk 2693 2597 a(2) p Fn 2787 2597 a(L) p Fl 2853 2556 a(2) p Fo 2893 2597 a(\() p Fj(R) p Fl 3018 2555 a(2) p Fo 3058 2597 a(\)) p Fk(g) p Fn(:) p Ff 146 2817 a(\(ii\)) p 323 2817 a(L) p 379 2817 a(et) p Fn 495 2817 a(u) p Fk 583 2817 a(2) p Fn 682 2817 a(D) p Fo 766 2817 a(\() p Fn(L) p Fg 870 2781 a(\003) p Fm 870 2842 a(N) p Fo 938 2817 a(\)) p Ff(.) p 1059 2817 a(Supp) p 1261 2817 a(ose) p 1433 2817 a(that) p 1635 2817 a(ther) p 1798 2817 a(e) p 1880 2817 a(exist) p 2110 2817 a(a) p 2197 2817 a(c) p 2237 2817 a(onstant) p Fn 2589 2817 a(R) p Fl 2663 2832 a(1) p Ff 2740 2817 a(with) p Fo 2955 2817 a(0) p Fn 3036 2817 a(<) p 3145 2817 a(R) p Fl 3219 2832 a(1) p Fn 3292 2817 a(<) p 3400 2817 a(R) p Ff 3513 2817 a(such) p 3738 2817 a(that) p Fo 0 2938 a(supp) p Fn 218 2938 a(u) p Fk 301 2938 a(\032) p Fj 406 2938 a(R) p Fl 493 2895 a(2) p Fk 555 2938 a(n) p Fn 627 2938 a(U) p Fo 703 2938 a(\() p Fn(R) p Fl 815 2953 a(1) p Fo 855 2938 a(\)) p Ff(.) p 968 2938 a(Then,) p Fn 1252 2938 a(u) p Fk 1335 2938 a(2) p Fn 1429 2938 a(D) p Fo 1513 2938 a(\() p 1551 2860 134 4 v Fn(L) p Fm 1617 2953 a(N) p Fo 1684 2938 a(\)) p Ff(.) 146 3166 y(Pr) p 248 3166 a(o) p 293 3166 a(of.) p Fo 444 3166 a(\(i\)) p 588 3166 a(By) p 749 3166 a(de\014nition,) p Fn 1220 3166 a(u) p Fk 1318 3166 a(2) p Fn 1426 3166 a(D) p Fo 1510 3166 a(\() p Fn(L) p Fg 1614 3130 a(\003) p Fm 1614 3191 a(N) p Fo 1681 3166 a(\)) p 1760 3166 a(if) p 1858 3166 a(and) p 2056 3166 a(only) p 2278 3166 a(if) p 2376 3166 a(there) p 2634 3166 a(exists) p 2911 3166 a(some) p Fn 3164 3166 a(v) p Fk 3257 3166 a(2) p Fn 3365 3166 a(L) p Fl 3431 3130 a(2) p Fo 3471 3166 a(\() p Fj(R) p Fl 3596 3124 a(2) p Fo 3636 3166 a(\)) p 3714 3166 a(suc) m(h) 0 3286 y(that) p 214 3286 a(\() p Fn(v) t(;) p 347 3286 a(\036) p Fo(\)) p 474 3286 a(=) p 582 3286 a(\() p Fn(u;) p Fk 720 3286 a(\000) p Fo(\() p Fk(r) p Fo 941 3286 a(+) p Fn 1041 3286 a(i) p Fj(a) p Fm 1137 3301 a(N) p Fo 1204 3286 a(\)) p Fl 1242 3250 a(2) p Fn 1282 3286 a(\036) p Fo(\)) p 1413 3286 a(for) p 1564 3286 a(ev) m(ery) p Fn 1824 3286 a(\036) p Fk 1914 3286 a(2) p Fn 2012 3286 a(C) p Fg 2089 3250 a(1) p Fl 2082 3311 a(0) p Fo 2164 3286 a(\() p Fj(R) p Fl 2289 3244 a(2) p Fk 2352 3286 a(n) p Fn 2426 3286 a(S) p Fm 2486 3301 a(N) p Fo 2553 3286 a(\).) p 2669 3286 a(By) p 2825 3286 a(de\014nition,) p 3289 3286 a(this) p 3481 3286 a(statemen) m(t) 0 3407 y(is) p 99 3407 a(equiv) p 321 3407 a(alen) m(t) p 563 3407 a(to) p Fn 682 3407 a(v) p Fo 762 3407 a(=) p Fk 866 3407 a(\000) p Fo(\() p Fk(r) p Fo 1087 3407 a(+) p Fn 1186 3407 a(i) p Fj(a) p Fm 1282 3422 a(N) p Fo 1349 3407 a(\)) p Fl 1387 3370 a(2) p Fn 1427 3407 a(u) p Fo 1515 3407 a(in) p Fk 1630 3407 a(D) p Fg 1710 3370 a(0) p Fo 1733 3407 a(\() p Fj(R) p Fl 1858 3365 a(2) p Fk 1920 3407 a(n) p Fn 1992 3407 a(S) p Fm 2052 3422 a(N) p Fo 2119 3407 a(\).) p 2229 3407 a(Then) p 2485 3407 a(w) m(e) p 2629 3407 a(ha) m(v) m(e) p Fn 2854 3407 a(u) p Fk 2939 3407 a(2) p Fn 3034 3407 a(H) p Fl 3123 3370 a(2) p Fm 3115 3431 a(l) q(oc) p Fo 3205 3407 a(\() p Fj(R) p Fl 3330 3365 a(2) p Fk 3392 3407 a(n) p Fn 3465 3407 a(S) p Fm 3525 3422 a(N) p Fo 3592 3407 a(\)) p 3663 3407 a(b) m(y) p 3799 3407 a(an) 0 3527 y(elliptic) p 320 3527 a(estimate) p 710 3527 a(\(see) p 906 3527 a([Ag]\).) 146 3647 y(\(ii\)) p 308 3647 a(T) p 370 3647 a(ak) m(e) p Fn 542 3647 a(\021) p Fk 622 3647 a(2) p Fn 716 3647 a(C) p Fg 793 3611 a(1) p Fo 868 3647 a(\() p Fj(R) p Fl 993 3605 a(2) p Fo 1032 3647 a(\)) p 1102 3647 a(satisfying) p Fn 1535 3647 a(\021) p Fo 1615 3647 a(=) p 1718 3647 a(1) p 1798 3647 a(on) p Fj 1933 3647 a(R) p Fl 2020 3605 a(2) p Fk 2079 3647 a(n) p Fn 2149 3647 a(U) p Fo 2225 3647 a(\() p Fl 2273 3608 a(1) p 2273 3624 36 4 v 2273 3682 a(3) p Fn 2319 3647 a(R) p Fl 2393 3662 a(1) p Fo 2433 3647 a(\),) p Fn 2529 3647 a(\021) p Fo 2609 3647 a(=) p 2712 3647 a(0) p 2792 3647 a(on) p Fn 2927 3647 a(U) p Fo 3003 3647 a(\() p Fl 3051 3608 a(1) p 3051 3624 V 3051 3682 a(4) p Fn 3097 3647 a(R) p Fl 3171 3662 a(1) p Fo 3210 3647 a(\)) p 3280 3647 a(and) p Fn 3468 3647 a(\021) p Fo 3520 3647 a(,) p Fk 3579 3647 a(r) p Fn(\021) p Fo 3745 3647 a(and) 0 3768 y(\001) p Fn(\021) p Fo 166 3768 a(are) p 328 3768 a(b) s(ounded) p 726 3768 a(functions.) p 1185 3768 a(Put) p Fi 1381 3759 a(g) p Fj 1376 3768 a(a) p Fm 1439 3783 a(N) p Fo 1534 3768 a(=) p Fn 1637 3768 a(\021) p Fj 1689 3768 a(a) p Fm 1752 3783 a(N) p Fo 1819 3768 a(.) p 1890 3768 a(Then) p 2145 3768 a(the) p 2313 3768 a(op) s(erator) p Fi 1139 3994 a(g) p Fn 1132 4027 a(L) p Fm 1198 4042 a(N) p Fo 1294 4027 a(=) p 1397 4027 a(\() 1445 3959 y(1) p 1445 4003 49 4 v Fn 1453 4095 a(i) p Fk 1504 4027 a(r) p Fo 1609 4027 a(+) p Fi 1712 4018 a(g) p Fj 1707 4027 a(a) p Fm 1770 4042 a(N) p Fo 1837 4027 a(\)) p Fl 1875 3986 a(2) p Fn 1915 4027 a(;) p 1991 4027 a(D) p Fo 2075 4027 a(\() p Fi 2120 3994 a(g) p Fn 2113 4027 a(L) p Fm 2179 4042 a(N) p Fo 2246 4027 a(\)) p 2312 4027 a(=) p Fn 2415 4027 a(C) p Fg 2492 3986 a(1) p Fl 2485 4051 a(0) p Fo 2567 4027 a(\() p Fj(R) p Fl 2692 3985 a(2) p Fo 2732 4027 a(\)) 0 4288 y(is) p 98 4288 a(essen) m(tially) p 565 4288 a(self-adjoin) m(t) p 1067 4288 a(since) p Fi 1311 4279 a(g) p Fj 1306 4288 a(a) p Fm 1369 4303 a(N) p Fk 1464 4288 a(2) p Fn 1558 4288 a(C) p Fg 1635 4252 a(1) p Fo 1710 4288 a(\() p Fj(R) p Fl 1835 4246 a(2) p Fo 1875 4288 a(\)) p 1945 4288 a(\(see) p 2141 4288 a([Ik-Ka]\).) p 2547 4288 a(Th) m(us) p 2794 4288 a(w) m(e) p 2938 4288 a(ha) m(v) m(e) p Fn 927 4532 a(D) p Fo 1011 4532 a(\() p 1049 4427 134 4 v Fi 1056 4500 a(g) p Fn 1049 4532 a(L) p Fm 1115 4547 a(N) p Fo 1182 4532 a(\)) p 1248 4532 a(=) p Fk 1351 4532 a(f) p Fn(u) p Fk 1484 4532 a(2) p Fn 1578 4532 a(L) p Fl 1644 4491 a(2) p Fo 1684 4532 a(\() p Fj(R) p Fl 1809 4490 a(2) p Fo 1849 4532 a(\);) p 1931 4532 a(\() p Fk(r) p Fo 2074 4532 a(+) p Fn 2172 4532 a(i) p Fi 2210 4523 a(g) p Fj 2205 4532 a(a) p Fm 2268 4547 a(N) p Fo 2335 4532 a(\)) p Fl 2373 4491 a(2) p Fn 2413 4532 a(u) p Fk 2496 4532 a(2) p Fn 2590 4532 a(L) p Fl 2656 4491 a(2) p Fo 2696 4532 a(\() p Fj(R) p Fl 2821 4490 a(2) p Fo 2860 4532 a(\)) p Fk(g) p Fn(;) p Fo 3701 4532 a(\(4.4\)) 0 4752 y(b) s(ecause) p 366 4752 a(the) p 539 4752 a(domain) p 891 4752 a(of) p 1007 4752 a(the) p 1179 4752 a(maximal) p 1583 4752 a(op) s(erator) p 1981 4752 a(coincides) p 2398 4752 a(with) p 2625 4752 a(that) p 2841 4752 a(of) p 2957 4752 a(the) p 3130 4752 a(minimal) p 3515 4752 a(op) s(erator.) 0 4891 y(Since) p Fi 262 4858 a(g) p Fn 255 4891 a(L) p Fm 321 4906 a(N) p Fn 388 4891 a(u) p Fo 472 4891 a(=) p Fk 575 4891 a(L) p Fm 644 4906 a(N) p Fn 711 4891 a(u) p Fk 794 4891 a(2) p Fn 888 4891 a(L) p Fl 954 4855 a(2) p Fo 994 4891 a(\() p Fj(R) p Fl 1119 4849 a(2) p Fo 1159 4891 a(\),) p 1256 4891 a(w) m(e) p 1400 4891 a(ha) m(v) m(e) p Fn 1625 4891 a(u) p Fk 1708 4891 a(2) p Fn 1802 4891 a(D) p Fo 1886 4891 a(\() p 1924 4786 V Fi 1931 4858 a(g) p Fn 1924 4891 a(L) p Fm 1990 4906 a(N) p Fo 2057 4891 a(\)) p 2128 4891 a(b) m(y) p 2263 4891 a(\(4.4\).) 1902 5214 y(19) p eop %%Page: 20 20 20 19 bop Fo 146 631 a(By) p 294 631 a(the) p 457 631 a(essen) m(tial) p 841 631 a(self-adjoin) m(tness) p 1512 631 a(of) p Fi 1625 599 a(g) p Fn 1618 631 a(L) p Fm 1684 646 a(N) p Fo 1752 631 a(,) p 1807 631 a(w) m(e) p 1946 631 a(can) p 2120 631 a(tak) m(e) p 2326 631 a(a) p 2402 631 a(sequence) p Fk 2802 631 a(f) p Fn(u) p Fm 2908 646 a(n) p Fk 2954 631 a(g) p Fg 3004 595 a(1) p Fm 3004 656 a(n) p Fl(=1) p Fk 3168 631 a(\032) p Fn 3274 631 a(C) p Fg 3351 595 a(1) p Fl 3344 656 a(0) p Fo 3425 631 a(\() p Fj(R) p Fl 3551 589 a(2) p Fo 3590 631 a(\)) p 3655 631 a(whic) m(h) 0 775 y(appro) m(ximates) p Fn 607 775 a(u) p Fo 700 775 a(with) p 927 775 a(resp) s(ect) p 1265 775 a(to) p 1390 775 a(the) p 1562 775 a(graph) p 1844 775 a(norm) p 2103 775 a(of) p 2219 671 134 4 v Fi 2226 743 a(g) p Fn 2219 775 a(L) p Fm 2285 790 a(N) p Fo 2353 775 a(.) p 2438 775 a(T) p 2500 775 a(ak) m(e) p Fn 2678 775 a(\030) p Fk 2761 775 a(2) p Fn 2864 775 a(C) p Fg 2941 739 a(1) p Fo 3015 775 a(\() p Fj(R) p Fl 3141 733 a(2) p Fo 3180 775 a(\),) p 3284 775 a(satisfying) p 3723 775 a(that) p Fn 0 896 a(\030) p Fo 76 896 a(=) p 181 896 a(1) p 263 896 a(on) p Fj 399 896 a(R) p Fl 486 854 a(2) p Fk 548 896 a(n) p Fn 621 896 a(U) p Fo 697 896 a(\() p Fm 745 857 a(R) p Fd 798 866 a(1) p 745 873 88 4 v Fl 771 930 a(2) p Fo 843 896 a(\),) p 941 896 a(that) p Fn 1153 896 a(\030) p Fo 1230 896 a(=) p 1334 896 a(0) p 1416 896 a(on) p Fn 1552 896 a(U) p Fo 1628 896 a(\() p Fm 1676 857 a(R) p Fd 1729 866 a(1) p 1677 873 V Fl 1703 930 a(3) p Fo 1774 896 a(\)) p 1846 896 a(and) p 2036 896 a(that) p Fn 2248 896 a(\030) p Fo 2296 896 a(,) p Fk 2356 896 a(r) p Fn(\030) p Fo 2487 896 a(,) p 2547 896 a(\001) p Fn(\030) p Fo 2709 896 a(are) p 2872 896 a(b) s(ounded.) p 3310 896 a(Then) p 3565 896 a(w) m(e) p 3710 896 a(ha) m(v) m(e) p Fk 0 1016 a(f) p Fn(\030) p 98 1016 a(u) p Fm 154 1031 a(n) p Fk 200 1016 a(g) p Fg 250 980 a(1) p Fm 250 1041 a(n) p Fl(=1) p Fk 424 1016 a(\032) p Fn 539 1016 a(D) p Fo 623 1016 a(\() p Fn(L) p Fm 727 1031 a(N) p Fo 794 1016 a(\).) p 920 1016 a(It) p 1031 1016 a(is) p 1135 1016 a(su\016cien) m(t) p 1550 1016 a(to) p 1675 1016 a(sho) m(w) p 1922 1016 a(that) p 2139 1016 a(the) p 2313 1016 a(sequence) p Fk 2723 1016 a(f) p Fn(\030) p 2821 1016 a(u) p Fm 2877 1031 a(n) p Fk 2923 1016 a(g) p Fo 3011 1016 a(appro) m(ximates) p Fn 3619 1016 a(u) p Fo 3712 1016 a(with) 0 1137 y(resp) s(ect) p 334 1137 a(to) p 453 1137 a(the) p 621 1137 a(graph) p 897 1137 a(norm) p 1152 1137 a(of) p Fn 1263 1137 a(L) p Fg 1329 1100 a(\003) p Fm 1329 1161 a(N) p Fo 1397 1137 a(.) p 1467 1137 a(Since) p 978 1258 134 4 v Fi 985 1330 a(g) p Fn 978 1363 a(L) p Fm 1044 1378 a(N) p Fo 1112 1363 a(\() p Fn(\030) p 1198 1363 a(\036) p Fo(\)) p 1320 1363 a(=) p Fn 1424 1363 a(\030) p 1472 1258 V Fi 1479 1330 a(g) p Fn 1472 1363 a(L) p Fm 1538 1378 a(N) p Fn 1605 1363 a(\036) p Fk 1685 1363 a(\000) p Fo 1784 1363 a(2\() p Fk(r) p Fn(\030) p Fo 2002 1363 a(\)\() p Fk(r) p Fo 2183 1363 a(+) p Fn 2281 1363 a(i) p Fi 2319 1354 a(g) p Fj 2314 1363 a(a) p Fm 2377 1378 a(N) p Fo 2444 1363 a(\)) p Fn(\036) p Fk 2562 1363 a(\000) p Fo 2662 1363 a(\(\001) p Fn(\030) p Fo 2829 1363 a(\)) p Fn(\036) p Fo 0 1589 a(for) p 149 1589 a(an) m(y) p Fn 333 1589 a(\036) p Fk 419 1589 a(2) p Fn 513 1589 a(D) p Fo 597 1589 a(\() p 635 1484 V Fi 642 1557 a(g) p Fn 635 1589 a(L) p Fm 701 1604 a(N) p Fo 768 1589 a(\),) p 866 1589 a(w) m(e) p 1009 1589 a(ha) m(v) m(e) p Fk 698 1815 a(jj) p Fn(L) p Fg 820 1774 a(\003) p Fm 820 1840 a(N) p Fo 887 1815 a(\() p Fn(\030) p 973 1815 a(u) p Fm 1029 1830 a(n) p Fk 1097 1815 a(\000) p Fn 1196 1815 a(u) p Fo(\)) p Fk(jj) p Fo 1373 1815 a(=) p Fk 1476 1815 a(jj) p 1532 1711 V Fi 1539 1783 a(g) p Fn 1532 1815 a(L) p Fm 1598 1830 a(N) p Fo 1665 1815 a(\() p Fn(\030) p Fo 1751 1815 a(\() p Fn(u) p Fm 1845 1830 a(n) p Fk 1913 1815 a(\000) p Fn 2013 1815 a(u) p Fo(\)\)) p Fk(jj) 537 1985 y(\024) p 698 1985 a(jj) p Fn(\030) p 802 1880 V Fi 809 1952 a(g) p Fn 802 1985 a(L) p Fm 868 2000 a(N) p Fo 934 1985 a(\() p Fn(u) p Fm 1028 2000 a(n) p Fk 1097 1985 a(\000) p Fn 1196 1985 a(u) p Fo(\)) p Fk(jj) p Fo 1367 1985 a(+) p 1465 1985 a(2) p Fk(jj) p Fo(\() p Fk(r) p Fn(\030) p Fo 1739 1985 a(\)\() p Fk(r) p Fo 1919 1985 a(+) p Fn 2017 1985 a(i) p Fi 2055 1976 a(g) p Fj 2050 1985 a(a) p Fm 2113 2000 a(N) p Fo 2181 1985 a(\)\() p Fn(u) p Fm 2313 2000 a(n) p Fk 2381 1985 a(\000) p Fn 2481 1985 a(u) p Fo(\)) p Fk(jj) p Fo 2652 1985 a(+) p Fk 2750 1985 a(jj) p Fo(\(\001) p Fn(\030) p Fo 2973 1985 a(\)\() p Fn(u) p Fm 3105 2000 a(n) p Fk 3172 1985 a(\000) p Fn 3272 1985 a(u) p Fo(\)) p Fk(jj) p Fn(:) p Fo 0 2187 a(One) p 215 2187 a(can) p 402 2187 a(sho) m(w) p 653 2187 a(that) p 873 2187 a(the) p 1049 2187 a(last) p 1243 2187 a(expression) p 1721 2187 a(con) m(v) m(erges) p 2167 2187 a(to) p 2295 2187 a(0) p 2385 2187 a(as) p Fn 2513 2187 a(n) p Fk 2614 2187 a(!) p 2756 2187 a(1) p Fo 2897 2187 a(b) m(y) p 3041 2187 a(using) p 3305 2187 a(the) p 3482 2187 a(inequalit) m(y) 0 2307 y(\(whic) m(h) p 317 2307 a(can) p 496 2307 a(b) s(e) p 629 2307 a(pro) m(v) m(ed) p 946 2307 a(b) m(y) p 1081 2307 a(an) p 1217 2307 a(in) m(tegration) p 1713 2307 a(b) m(y) p 1848 2307 a(parts\)) p Fk 969 2554 a(jj) p Fo(\() p Fk(r) p Fo 1167 2554 a(+) p Fn 1265 2554 a(i) p Fi 1303 2545 a(g) p Fj 1298 2554 a(a) p Fm 1361 2569 a(N) p Fo 1429 2554 a(\)) p Fn(v) p Fk 1518 2554 a(jj) p Fl 1574 2513 a(2) p Fk 1640 2554 a(\024) p Fo 1755 2487 a(1) p 1755 2531 49 4 v 1755 2623 a(2) p Fi 1830 2458 a(\020) p Fk 1880 2554 a(jj) p Fo(\() p Fk(r) p Fo 2078 2554 a(+) p Fn 2176 2554 a(i) p Fi 2214 2545 a(g) p Fj 2209 2554 a(a) p Fm 2273 2569 a(N) p Fo 2340 2554 a(\)) p Fl 2378 2513 a(2) p Fn 2417 2554 a(v) p Fk 2468 2554 a(jj) p Fl 2524 2513 a(2) p Fo 2585 2554 a(+) p Fk 2683 2554 a(jj) p Fn(v) p Fk 2790 2554 a(jj) p Fl 2846 2513 a(2) p Fi 2884 2458 a(\021) p Fo 0 2824 a(for) p Fn 149 2824 a(v) p Fk 227 2824 a(2) p Fn 322 2824 a(D) p Fo 406 2824 a(\() p 444 2719 134 4 v Fi 451 2792 a(g) p Fn 444 2824 a(L) p Fm 510 2839 a(N) p Fo 577 2824 a(\)) p 647 2824 a(and) p 837 2824 a(the) p 1005 2824 a(uniform) p 1371 2824 a(b) s(oundedness) p 1944 2824 a(of) p Fk 2055 2824 a(jr) p Fn(\030) p Fk 2214 2824 a(j) p Fo(,) p Fk 2300 2824 a(j) p Fo(\001) p Fn(\030) p Fk 2457 2824 a(j) p Fo(.) p 2555 2756 64 4 v 2555 2849 4 93 v 2615 2849 V 2555 2852 64 4 v 146 3032 a(W) p 238 3032 a(e) p 314 3032 a(shall) p 541 3032 a(pro) m(v) m(e) p 804 3032 a(the) p 971 3032 a(con) m(tin) m(uit) m(y) p 1431 3032 a(of) p 1542 3032 a(the) p 1709 3032 a(cutting-o\013) p 2183 3032 a(map.) p 2438 3032 a(In) p 2559 3032 a(the) p 2726 3032 a(sequel,) p 3044 3032 a(w) m(e) p 3187 3032 a(regard) p Fn 3490 3032 a(D) p Fo 3574 3032 a(\() p Fn(L) p Fg 3678 2996 a(\003) p Fm 3678 3057 a(N) p Fo 3745 3032 a(\)) p 3815 3032 a(as) 0 3152 y(a) p 81 3152 a(Hilb) s(ert) p 417 3152 a(space) p 678 3152 a(equipp) s(ed) p 1095 3152 a(with) p 1317 3152 a(the) p 1485 3152 a(graph) p 1762 3152 a(norm) p Fk 1148 3355 a(jj) p Fn(u) p Fk(jj) p Fl 1316 3314 a(2) p Fm 1316 3379 a(N) p Fo 1409 3355 a(=) p Fk 1513 3355 a(jjL) p Fm 1638 3370 a(N) p Fn 1704 3355 a(u) p Fk(jj) p Fl 1816 3314 a(2) p Fo 1876 3355 a(+) p Fk 1974 3355 a(jj) p Fn(u) p Fk(jj) p Fl 2142 3314 a(2) p Fn 2180 3355 a(;) p 2256 3355 a(u) p Fk 2340 3355 a(2) p Fn 2434 3355 a(D) p Fo 2518 3355 a(\() p Fn(L) p Fg 2622 3314 a(\003) p Fm 2622 3379 a(N) p Fo 2689 3355 a(\)) p Fn(:) p Fo 0 3557 a(The) p 201 3557 a(brac) m(k) m(et) p 546 3557 a([) p Fn(X) r(;) p 700 3557 a(Y) p Fo 778 3557 a(]) p 838 3557 a(denotes) p 1192 3557 a(the) p 1360 3557 a(comm) m(utator) p 1911 3557 a(of) p 2022 3557 a(t) m(w) m(o) p 2207 3557 a(op) s(erators) p Fn 2639 3557 a(X) r(;) p 2766 3557 a(Y) p Fo 2844 3557 a(,) p 2905 3557 a(that) p 3117 3557 a(is,) p 3242 3557 a([) p Fn(X) r(;) p 3396 3557 a(Y) p Fo 3475 3557 a(]) p 3531 3557 a(=) p Fn 3635 3557 a(X) p 3724 3557 a(Y) p Fk 3825 3557 a(\000) p Fn 0 3678 a(Y) p 78 3678 a(X) p Fo 167 3678 a(.) p Fq 146 3866 a(Lemma) p 543 3866 a(4.3) p Ff 735 3866 a(L) p 791 3866 a(et) p Fo 906 3866 a(1) p Fk 987 3866 a(\024) p Fn 1097 3866 a(N) p 1218 3866 a(<) p Fk 1327 3866 a(1) p Ff 1464 3866 a(or) p Fn 1592 3866 a(N) p Fo 1713 3866 a(=) p Fk 1821 3866 a(1) p Ff(.) p 2004 3866 a(L) p 2060 3866 a(et) p Fn 2174 3866 a(\021) p Fk 2259 3866 a(2) p Fn 2357 3866 a(C) p Fg 2434 3830 a(1) p Fo 2509 3866 a(\() p Fj(R) p Fl 2634 3824 a(2) p Fo 2674 3866 a(\)) p Ff 2749 3866 a(and) p 2941 3866 a(supp) p 3128 3866 a(ose) p Fo 3300 3866 a(supp) p Fk 3517 3866 a(jr) p Fn(\021) p Fk 3680 3866 a(j) p Ff 3745 3866 a(is) p 3852 3866 a(a) 0 3986 y(c) p 40 3986 a(omp) p 215 3986 a(act) p 376 3986 a(set) p 528 3986 a(in) p Fj 648 3986 a(R) p Fl 735 3944 a(2) p Fk 796 3986 a(n) p Fn 868 3986 a(S) p Fm 928 4001 a(N) p Ff 996 3986 a(.) p 1070 3986 a(Then,) p 1354 3986 a(for) p 1510 3986 a(any) p Fn 1697 3986 a(u) p Fk 1780 3986 a(2) p Fn 1874 3986 a(D) p Fo 1958 3986 a(\() p Fn(L) p Fg 2062 3950 a(\003) p Fm 2062 4011 a(N) p Fo 2129 3986 a(\)) p Ff(,) p 2232 3986 a(we) p 2376 3986 a(have) p Fn 2601 3986 a(\021) t(u) p Fk 2736 3986 a(2) p Fn 2830 3986 a(D) p Fo 2914 3986 a(\() p Fn(L) p Fg 3018 3950 a(\003) p Fm 3018 4011 a(N) p Fo 3085 3986 a(\)) p Ff 3158 3986 a(and) p Fk 971 4219 a(jj) p Fo([) p Fk(L) p Fm 1123 4234 a(N) p Fn 1189 4219 a(;) p 1233 4219 a(\021) p Fo 1285 4219 a(]) p Fn(u) p Fk(jj) p Fl 1424 4177 a(2) p Fk 1489 4219 a(\024) p Fn 1595 4219 a(C) p Fl 1665 4234 a(0) p Fi 1720 4101 a(Z) p Fl 1767 4290 a(supp) p Fg 1924 4290 a(jr) p Fm(\021) p Fg 2040 4290 a(j) p Fo 2064 4219 a(\() p Fk(jL) p Fm 2199 4234 a(N) p Fn 2265 4219 a(u) p Fk(j) p Fl 2349 4177 a(2) p Fo 2410 4219 a(+) p Fk 2508 4219 a(j) p Fn(u) p Fk(j) p Fl 2620 4177 a(2) p Fo 2658 4219 a(\)) p Fn(dxdy) t(;) p Fo 3701 4219 a(\(4.5\)) p Ff 0 4476 a(wher) p 196 4476 a(e) p Fn 275 4476 a(C) p Fl 345 4491 a(0) p Fo 412 4476 a(=) p 516 4476 a(10) p 631 4476 a(sup\() p Fk(jr) p Fn(\021) p Fk 978 4476 a(j) p Fl 1006 4440 a(2) p Fo 1066 4476 a(+) p Fk 1164 4476 a(jr) p Fo(\() p Fn(@) p Fm 1364 4491 a(x) p Fn 1408 4476 a(\021) p Fo 1460 4476 a(\)) p Fk(j) p Fl 1526 4440 a(2) p Fo 1587 4476 a(+) p Fk 1685 4476 a(jr) p Fo(\() p Fn(@) p Fm 1885 4491 a(y) p Fn 1927 4476 a(\021) p Fo 1979 4476 a(\)) p Fk(j) p Fl 2045 4440 a(2) p Fo 2083 4476 a(\)) p Ff(.) p 2196 4476 a(Mor) p 2369 4476 a(e) p 2409 4476 a(over,) p 2655 4476 a(we) p 2799 4476 a(have) p Fk 1123 4711 a(jj) p Fn(\021) t(u) p Fk(jj) p Fl 1343 4670 a(2) p Fm 1343 4736 a(N) p Fk 1435 4711 a(\024) p Fn 1541 4711 a(C) p Fl 1611 4726 a(1) p Fi 1666 4594 a(Z) p Fl 1713 4783 a(supp) p Fm 1870 4783 a(\021) p Fo 1912 4711 a(\() p Fk(jL) p Fm 2047 4726 a(N) p Fn 2113 4711 a(u) p Fk(j) p Fl 2197 4670 a(2) p Fo 2258 4711 a(+) p Fk 2356 4711 a(j) p Fn(u) p Fk(j) p Fl 2468 4670 a(2) p Fo 2506 4711 a(\)) p Fn(dxdy) t(;) p Fo 3701 4711 a(\(4.6\)) p Ff 0 4965 a(wher) p 196 4965 a(e) p Fn 275 4965 a(C) p Fl 345 4980 a(1) p Fo 412 4965 a(=) p 516 4965 a(2\(sup) p Fk 766 4965 a(j) p Fn(\021) p Fk 846 4965 a(j) p Fl 874 4929 a(2) p Fo 934 4965 a(+) p Fn 1032 4965 a(C) p Fl 1102 4980 a(0) p Fo 1142 4965 a(\)) p Ff(.) p Fo 1902 5214 a(20) p eop %%Page: 21 21 21 20 bop Ff 146 631 a(Pr) p 248 631 a(o) p 293 631 a(of.) p Fo 445 631 a(T) p 507 631 a(ak) m(e) p Fn 690 631 a(u) p Fk 790 631 a(2) p Fn 901 631 a(D) p Fo 985 631 a(\() p Fn(L) p Fg 1089 595 a(\003) p Fm 1089 656 a(N) p Fo 1156 631 a(\).) p 1293 631 a(By) p 1456 631 a(\(i\)) p 1601 631 a(of) p 1722 631 a(Prop) s(osition) p 2256 631 a(4.2,) p 2452 631 a(w) m(e) p 2606 631 a(ha) m(v) m(e) p Fk 2840 631 a(L) p Fm 2909 646 a(N) p Fn 2976 631 a(u) p Fk 3076 631 a(2) p Fn 3186 631 a(L) p Fl 3252 595 a(2) p Fo 3292 631 a(\() p Fj(R) p Fl 3417 589 a(2) p Fo 3457 631 a(\)) p 3537 631 a(and) p Fn 3736 631 a(u) p Fk 3836 631 a(2) p Fn 0 751 a(H) p Fl 89 715 a(2) p Fm 81 776 a(l) q(oc) p Fo 172 751 a(\() p Fj(R) p Fl 297 709 a(2) p Fk 358 751 a(n) p Fn 430 751 a(S) p Fm 490 766 a(N) p Fo 558 751 a(\).) p 666 751 a(Then) p Fn 921 751 a(\021) t(u) p Fk 1056 751 a(2) p Fn 1150 751 a(L) p Fl 1216 715 a(2) p Fo 1255 751 a(\() p Fj(R) p Fl 1381 709 a(2) p Fo 1420 751 a(\)) p 1491 751 a(and) p Fk 743 971 a(L) p Fm 812 986 a(N) p Fo 879 971 a(\() p Fn(\021) t(u) p Fo(\)) p 1090 971 a(=) p Fn 1193 971 a(\021) p Fk 1245 971 a(L) p Fm 1314 986 a(N) p Fn 1381 971 a(u) p Fk 1458 971 a(\000) p Fo 1558 971 a(2\() p Fk(r) p Fn(\021) p Fo 1780 971 a(\)\() p Fk(r) p Fo 1961 971 a(+) p Fn 2059 971 a(i) p Fj(a) p Fm 2155 986 a(N) p Fo 2222 971 a(\)) p Fn(u) p Fk 2338 971 a(\000) p Fo 2437 971 a(\(\001) p Fn(\021) p Fo 2608 971 a(\)) p Fn(u) p Fk 2729 971 a(2) p Fn 2824 971 a(L) p Fl 2890 930 a(2) p Fo 2929 971 a(\() p Fj(R) p Fl 3055 929 a(2) p Fo 3094 971 a(\)) p Fn(;) p Fo 0 1191 a(since) p 239 1191 a(supp) p Fk 457 1191 a(r) p Fn(\021) p Fo 624 1191 a(is) p 722 1191 a(compact.) p 1150 1191 a(Th) m(us) p Fn 1397 1191 a(\021) t(u) p Fk 1532 1191 a(2) p Fn 1626 1191 a(D) p Fo 1710 1191 a(\() p Fn(L) p Fg 1814 1155 a(\003) p Fm 1814 1216 a(N) p Fo 1881 1191 a(\)) p 1952 1191 a(b) m(y) p 2087 1191 a(\(i\)) p 2223 1191 a(of) p 2334 1191 a(Prop) s(osition) p 2859 1191 a(4.2.) 146 1312 y(Let) p Fn 314 1312 a(\030) p Fk 389 1312 a(2) p Fn 483 1312 a(C) p Fg 560 1276 a(1) p Fl 553 1336 a(0) p Fo 635 1312 a(\() p Fj(R) p Fl 760 1270 a(2) p Fk 807 1312 a(n) p Fn 865 1312 a(S) p Fm 925 1327 a(N) p Fo 992 1312 a(;) p Fj 1036 1312 a(R) p Fo(\).) p 1229 1312 a(The) p 1423 1312 a(follo) m(wing) p 1827 1312 a(equalit) m(y) p 2192 1312 a(is) p 2283 1312 a(a) p 2357 1312 a(little) p 2583 1312 a(mo) s(di\014ed) p 2974 1312 a(v) m(ersion) p 3298 1312 a(of) p 3402 1312 a(the) p 3563 1312 a(equalit) m(y) 0 1432 y(giv) m(en) p 255 1432 a(in) p 369 1432 a([Co-Ne,) p 723 1432 a(Lemma) p 1072 1432 a(I) s(I.2]:) p Fk 694 1652 a(jj) p Fo(\() p Fk(r) p Fo 892 1652 a(+) p Fn 990 1652 a(i) p Fj(a) p Fm 1086 1667 a(N) p Fo 1154 1652 a(\)\() p Fn(\030) p 1278 1652 a(u) p Fo(\)) p Fk(jj) p Fl 1428 1611 a(2) p Fo 1493 1652 a(=) p Fk 1596 1652 a(\000) p Fo 1690 1652 a(Re) q(\() p Fn(u;) p 1944 1652 a(\030) p Fl 1992 1611 a(2) p Fo 2030 1652 a(\() p Fk(r) p Fo 2173 1652 a(+) p Fn 2271 1652 a(i) p Fj(a) p Fm 2367 1667 a(N) p Fo 2434 1652 a(\)) p Fl 2472 1611 a(2) p Fn 2512 1652 a(u) p Fo(\)) p 2627 1652 a(+) p 2725 1652 a(\() p Fn(u;) p Fk 2863 1652 a(jr) p Fn(\030) p Fk 3022 1652 a(j) p Fl 3050 1611 a(2) p Fn 3088 1652 a(u) p Fo(\)) p Fn(:) p Fo 3701 1652 a(\(4.7\)) 0 1872 y(W) p 92 1872 a(e) p 168 1872 a(shall) p 396 1872 a(c) m(hec) m(k) p 659 1872 a(that) p 870 1872 a(\(4.7\)) p 1103 1872 a(holds) p 1359 1872 a(in) p 1472 1872 a(our) p 1646 1872 a(case.) p 1890 1872 a(By) p 2043 1872 a(the) p 2211 1872 a(Leibniz) p 2554 1872 a(rule,) p 2776 1872 a(w) m(e) p 2920 1872 a(ha) m(v) m(e) 831 2092 y(\(l) p Fn(:) p Fo(h) p Fn(:) p Fo(s) p Fn(:) p Fo 1102 2092 a(of) p 1220 2092 a(\(4) p Fn(:) p Fo(7\)\)) 672 2237 y(=) p Fk 831 2237 a(jj) p Fn(\030) p Fo 935 2237 a(\() p Fk(r) p Fo 1077 2237 a(+) p Fn 1175 2237 a(i) p Fj(a) p Fm 1271 2252 a(N) p Fo 1338 2237 a(\)) p Fn(u) p Fk(jj) p Fl 1488 2196 a(2) p Fo 1549 2237 a(+) p 1647 2237 a(2) p 1713 2237 a(Re) o(\(\() p Fk(r) p Fn(\030) p Fo 2034 2237 a(\)) p Fn(u;) p 2172 2237 a(\030) p Fo 2220 2237 a(\() p Fk(r) p Fo 2361 2237 a(+) p Fn 2459 2237 a(i) p Fj(a) p Fm 2555 2252 a(N) p Fo 2623 2237 a(\)) p Fn(u) p Fo(\)) p 2776 2237 a(+) p Fk 2874 2237 a(jj) p Fo(\() p Fk(r) p Fn(\030) p Fo 3099 2237 a(\)) p Fn(u) p Fk(jj) p Fl 3249 2196 a(2) p Fn 3286 2237 a(:) p Fo 3701 2237 a(\(4.8\)) 0 2457 y(Since) p Fn 464 2677 a(\030) p Fl 512 2636 a(2) p Fo 551 2677 a(\() p Fk(r) p Fo 694 2677 a(+) p Fn 792 2677 a(i) p Fj(a) p Fm 889 2692 a(N) p Fo 956 2677 a(\)) p Fl 994 2636 a(2) p Fo 1116 2677 a(=) p 1275 2677 a([) p Fn(\030) p Fl 1350 2636 a(2) p Fn 1389 2677 a(;) p Fo 1433 2677 a(\() p Fk(r) p Fo 1576 2677 a(+) p Fn 1674 2677 a(i) p Fj(a) p Fm 1770 2692 a(N) p Fo 1838 2677 a(\)]\() p Fk(r) p Fo 2046 2677 a(+) p Fn 2144 2677 a(i) p Fj(a) p Fm 2240 2692 a(N) p Fo 2307 2677 a(\)) p 2367 2677 a(+) p 2465 2677 a(\() p Fk(r) p Fo 2609 2677 a(+) p Fn 2707 2677 a(i) p Fj(a) p Fm 2803 2692 a(N) p Fo 2870 2677 a(\)) p Fn(\030) p Fl 2956 2636 a(2) p Fo 2995 2677 a(\() p Fk(r) p Fo 3138 2677 a(+) p Fn 3236 2677 a(i) p Fj(a) p Fm 3332 2692 a(N) p Fo 3400 2677 a(\)) 1116 2823 y(=) p Fk 1275 2823 a(\000) p Fo(2) p Fn(\030) p Fo 1449 2823 a(\() p Fk(r) p Fn(\030) p Fo 1618 2823 a(\)\() p Fk(r) p Fo 1798 2823 a(+) p Fn 1896 2823 a(i) p Fj(a) p Fm 1993 2838 a(N) p Fo 2060 2823 a(\)) p 2120 2823 a(+) p 2218 2823 a(\() p Fk(r) p Fo 2361 2823 a(+) p Fn 2459 2823 a(i) p Fj(a) p Fm 2556 2838 a(N) p Fo 2623 2823 a(\)) p Fn(\030) p Fl 2709 2782 a(2) p Fo 2748 2823 a(\() p Fk(r) p Fo 2891 2823 a(+) p Fn 2989 2823 a(i) p Fj(a) p Fm 3085 2838 a(N) p Fo 3152 2823 a(\)) p Fn(;) p Fo 0 3043 a(w) m(e) p 144 3043 a(ha) m(v) m(e) 510 3263 y(\(r) p Fn(:) p Fo(h) p Fn(:) p Fo(s) p Fn(:) p Fo 793 3263 a(of) p 911 3263 a(\(4) p Fn(:) p Fo(7\)) o(\)) 352 3408 y(=) p 510 3408 a(2) p 576 3408 a(Re\() p Fn(u;) p 829 3408 a(\030) p Fo 877 3408 a(\() p Fk(r) p Fn(\030) p Fo 1046 3408 a(\)\() p Fk(r) p Fo 1225 3408 a(+) p Fn 1323 3408 a(i) p Fj(a) p Fm 1419 3423 a(N) p Fo 1487 3408 a(\)) p Fn(u) p Fo(\)) p Fk 1640 3408 a(\000) p Fo 1740 3408 a(Re\() p Fn(u;) p Fo 1993 3408 a(\() p Fk(r) p Fo 2135 3408 a(+) p Fn 2233 3408 a(i) p Fj(a) p Fm 2329 3423 a(N) p Fo 2397 3408 a(\)) p Fn(\030) p Fl 2483 3367 a(2) p Fo 2521 3408 a(\() p Fk(r) p Fo 2665 3408 a(+) p Fn 2763 3408 a(i) p Fj(a) p Fm 2859 3423 a(N) p Fo 2926 3408 a(\)) p Fn(u) p Fo(\)) p 3080 3408 a(+) p 3178 3408 a(\() p Fn(u;) p Fk 3316 3408 a(jr) p Fn(\030) p Fk 3475 3408 a(j) p Fl 3503 3367 a(2) p Fn 3540 3408 a(u) p Fo(\)) 352 3553 y(=) p 510 3553 a(2) p 576 3553 a(Re\() p Fn(u;) p 829 3553 a(\030) p Fo 877 3553 a(\() p Fk(r) p Fn(\030) p Fo 1046 3553 a(\)\() p Fk(r) p Fo 1225 3553 a(+) p Fn 1323 3553 a(i) p Fj(a) p Fm 1419 3568 a(N) p Fo 1487 3553 a(\)) p Fn(u) p Fo(\)) p 1640 3553 a(+) p 1738 3553 a(Re\(\() p Fk(r) p Fo 2034 3553 a(+) p Fn 2132 3553 a(i) p Fj(a) p Fm 2229 3568 a(N) p Fo 2296 3553 a(\)) p Fn(u;) p 2434 3553 a(\030) p Fl 2482 3512 a(2) p Fo 2520 3553 a(\() p Fk(r) p Fo 2663 3553 a(+) p Fn 2761 3553 a(i) p Fj(a) p Fm 2857 3568 a(N) p Fo 2924 3553 a(\)) p Fn(u) p Fo(\)) p 3078 3553 a(+) p Fk 3176 3553 a(jj) p Fo(\() p Fk(r) p Fn(\030) p Fo 3401 3553 a(\)) p Fn(u) p Fk(jj) p Fl 3551 3512 a(2) p Fn 3588 3553 a(;) p Fo 0 3773 a(b) m(y) p 149 3773 a(partial) p 476 3773 a(in) m(tegration) p 985 3773 a(\(since) p Fn 1275 3773 a(\030) p Fk 1372 3773 a(2) p Fn 1489 3773 a(C) p Fg 1566 3737 a(1) p Fl 1559 3798 a(0) p Fo 1641 3773 a(\() p Fj(R) p Fl 1766 3731 a(2) p Fk 1836 3773 a(n) p Fn 1917 3773 a(S) p Fm 1977 3788 a(N) p Fo 2044 3773 a(\),) p 2158 3773 a(the) p 2339 3773 a(b) s(oundary) p 2792 3773 a(term) p 3038 3773 a(v) p 3084 3773 a(anishes\).) p 3536 3773 a(The) p 3750 3773 a(last) 0 3894 y(expression) p 470 3894 a(coincides) p 882 3894 a(with) p 1104 3894 a(\(4.8\).) p 1375 3894 a(Th) m(us) p 1623 3894 a(\(4.7\)) p 1856 3894 a(holds.) 146 4014 y(By) p 299 4014 a(\(4.7\)) p 532 4014 a(and) p 722 4014 a(the) p 890 4014 a(Sc) m(h) m(w) m(arz) p 1267 4014 a(inequalit) m(y) p 1680 4014 a(,) p 1738 4014 a(w) m(e) p 1882 4014 a(ha) m(v) m(e) p Fk 595 4264 a(jj) p Fo(\() p Fk(r) p Fo 793 4264 a(+) p Fn 891 4264 a(i) p Fj(a) p Fm 987 4279 a(N) p Fo 1055 4264 a(\)\() p Fn(\030) p 1179 4264 a(u) p Fo(\)) p Fk(jj) p Fl 1329 4223 a(2) p Fk 1394 4264 a(\024) p Fo 1499 4264 a(sup) q(\() p Fk(j) p Fn(\030) p Fk 1760 4264 a(j) p Fl 1788 4223 a(2) p Fo 1848 4264 a(+) p Fk 1946 4264 a(jr) p Fn(\030) p Fk 2105 4264 a(j) p Fl 2133 4223 a(2) p Fo 2171 4264 a(\)) p Fi 2226 4147 a(Z) p Fl 2272 4336 a(supp) p Fm 2429 4336 a(\030) p Fo 2467 4264 a(\() p Fk(jL) p Fm 2602 4279 a(N) p Fn 2668 4264 a(u) p Fk(j) p Fl 2752 4223 a(2) p Fo 2813 4264 a(+) p Fk 2911 4264 a(j) p Fn(u) p Fk(j) p Fl 3023 4223 a(2) p Fo 3062 4264 a(\)) p Fn(dxdy) p Fo 0 4541 a(for) p 149 4541 a(an) m(y) p Fn 333 4541 a(\030) p Fk 408 4541 a(2) p Fn 502 4541 a(C) p Fg 579 4505 a(1) p Fl 572 4566 a(0) p Fo 654 4541 a(\() p Fj(R) p Fl 779 4499 a(2) p Fk 841 4541 a(n) p Fn 913 4541 a(S) p Fm 973 4556 a(N) p Fo 1040 4541 a(\).) p 1149 4541 a(Applying) p 1571 4541 a(this) p 1761 4541 a(inequalit) m(y) p 2214 4541 a(to) p Fn 2333 4541 a(\030) p Fo 2408 4541 a(=) p Fn 2512 4541 a(@) p Fm 2563 4556 a(x) p Fn 2607 4541 a(\021) t(;) p 2703 4541 a(@) p Fm 2754 4556 a(y) p Fn 2796 4541 a(\021) p Fo 2848 4541 a(,) p 2907 4541 a(w) m(e) p 3050 4541 a(ha) m(v) m(e) p Fk 753 4794 a(jj) p Fo(\() p Fk(r) p Fo 952 4794 a(+) p Fn 1050 4794 a(i) p Fj(a) p Fm 1146 4809 a(N) p Fo 1213 4794 a(\)) p Fk(f) p Fo(\() p Fk(r) p Fn(\021) p Fo 1474 4794 a(\)) p Fn(u) p Fk(gjj) p Fl 1674 4753 a(2) p Fk 1739 4794 a(\024) p Fn 1844 4794 a(C) p Fi 1938 4677 a(Z) p Fl 1984 4866 a(supp) p Fg 2141 4866 a(jr) p Fm(\021) p Fg 2257 4866 a(j) p Fo 2281 4794 a(\() p Fk(jL) p Fm 2416 4809 a(N) p Fn 2483 4794 a(u) p Fk(j) p Fl 2567 4753 a(2) p Fo 2628 4794 a(+) p Fk 2726 4794 a(j) p Fn(u) p Fk(j) p Fl 2838 4753 a(2) p Fo 2876 4794 a(\)) p Fn(dxdy) t(;) p Fo 3701 4794 a(\(4.9\)) 1902 5214 y(21) p eop %%Page: 22 22 22 21 bop Fo 0 631 a(where) p Fn 282 631 a(C) p Fo 386 631 a(=) p 490 631 a(sup) q(\() p Fk(jr) p Fn(\021) p Fk 838 631 a(j) p Fl 866 595 a(2) p Fo 926 631 a(+) p Fk 1024 631 a(jr) p Fo(\() p Fn(@) p Fm 1224 646 a(x) p Fn 1268 631 a(\021) p Fo 1320 631 a(\)) p Fk(j) p Fl 1386 595 a(2) p Fo 1447 631 a(+) p Fk 1545 631 a(jr) p Fo(\() p Fn(@) p Fm 1745 646 a(y) p Fn 1787 631 a(\021) p Fo 1839 631 a(\)) p Fk(j) p Fl 1905 595 a(2) p Fo 1944 631 a(\).) p 2052 631 a(Since) 1044 834 y([) p Fk(L) p Fm 1140 849 a(N) p Fn 1208 834 a(;) p 1252 834 a(\021) p Fo 1304 834 a(]) p Fn(u) p Fo 1413 834 a(=) p Fk 1517 834 a(\000) p Fo(2\() p Fk(r) p Fo 1786 834 a(+) p Fn 1884 834 a(i) p Fj(a) p Fm 1980 849 a(N) p Fo 2048 834 a(\)) p 2103 834 a(\() o(\() p Fk(r) p Fn(\021) p Fo 2313 834 a(\)) p Fn(u) p Fo(\)) p Fk 2466 834 a(\000) p Fo 2566 834 a(\(\001) p Fn(\021) p Fo 2737 834 a(\)) p Fn(u;) p Fo 0 1037 a(w) m(e) p 144 1037 a(ha) m(v) m(e) p 369 1037 a(b) m(y) p 504 1037 a(\(4.9\)) p Fk 1273 1241 a(jj) p Fo([) p Fk(L) p Fm 1425 1256 a(N) p Fn 1491 1241 a(;) p 1535 1241 a(\021) p Fo 1587 1241 a(]) p Fn(u) p Fk(jj) p Fl 1726 1199 a(2) p Fk 1112 1386 a(\024) p Fo 1273 1386 a(8) p Fk(jj) p Fo(\() p Fk(r) p Fo 1520 1386 a(+) p Fn 1618 1386 a(i) p Fj(a) p Fm 1714 1401 a(N) p Fo 1781 1386 a(\)) p 1836 1386 a(\(\() p Fk(r) p Fn(\021) p Fo 2047 1386 a(\)) p Fn(u) p Fo(\)) p Fk 2195 1386 a(jj) p Fl 2251 1345 a(2) p Fo 2312 1386 a(+) p 2410 1386 a(2) p Fk(jj) p Fo(\(\001) p Fn(\021) p Fo 2686 1386 a(\)) p Fn(u) p Fk(jj) p Fl 2836 1345 a(2) p Fk 1112 1561 a(\024) p Fn 1273 1561 a(C) p Fl 1343 1576 a(0) p Fi 1399 1444 a(Z) p Fl 1445 1633 a(supp) p Fg 1602 1633 a(jr) p Fm(\021) p Fg 1718 1633 a(j) p Fo 1742 1561 a(\() p Fk(jL) p Fm 1877 1576 a(N) p Fn 1943 1561 a(u) p Fk(j) p Fl 2027 1520 a(2) p Fo 2088 1561 a(+) p Fk 2186 1561 a(j) p Fn(u) p Fk(j) p Fl 2298 1520 a(2) p Fo 2336 1561 a(\)) p Fn(dxdy) t(:) p Fo 0 1815 a(Th) m(us) p 247 1815 a(\(4.5\)) p 480 1815 a(holds.) p 773 1815 a(Moreo) m(v) m(er,) p 1229 1815 a(b) m(y) p 1364 1815 a(this) p 1554 1815 a(inequalit) m(y) p 2007 1815 a(w) m(e) p 2150 1815 a(ha) m(v) m(e) p Fk 1333 2018 a(jj) p Fn(\021) t(u) p Fk(jj) p Fl 1553 1977 a(2) p Fm 1553 2043 a(N) p Fo 1646 2018 a(=) p Fk 1749 2018 a(jjL) p Fm 1874 2033 a(N) p Fo 1941 2018 a(\() p Fn(\021) t(u) p Fo(\)) p Fk(jj) p Fl 2181 1977 a(2) p Fo 2241 2018 a(+) p Fk 2339 2018 a(jj) p Fn(\021) t(u) p Fk(jj) p Fl 2559 1977 a(2) p Fk 1172 2163 a(\024) p Fo 1333 2163 a(2) p Fk(jj) p Fo([) p Fk(L) p Fm 1534 2178 a(N) p Fn 1600 2163 a(;) p 1644 2163 a(\021) p Fo 1696 2163 a(]) p Fn(u) p Fk(jj) p Fl 1835 2122 a(2) p Fo 1895 2163 a(+) p 1993 2163 a(2) p Fk(jj) p Fn(\021) p Fk 2150 2163 a(L) p Fm 2219 2178 a(N) p Fn 2285 2163 a(u) p Fk(jj) p Fl 2397 2122 a(2) p Fo 2457 2163 a(+) p Fk 2555 2163 a(jj) p Fn(\021) t(u) p Fk(jj) p Fl 2775 2122 a(2) p Fk 1172 2339 a(\024) p Fn 1333 2339 a(C) p Fl 1403 2354 a(1) p Fi 1459 2222 a(Z) p Fl 1505 2410 a(supp) p Fm 1662 2410 a(\021) p Fo 1704 2339 a(\() p Fk(jL) p Fm 1839 2354 a(N) p Fn 1906 2339 a(u) p Fk(j) p Fl 1990 2298 a(2) p Fo 2050 2339 a(+) p Fk 2148 2339 a(j) p Fn(u) p Fk(j) p Fl 2260 2298 a(2) p Fo 2299 2339 a(\)) p Fn(dxdy) t(:) p Fo 0 2589 a(Th) m(us) p 247 2589 a(\(4.6\)) p 480 2589 a(holds.) p 773 2521 64 4 v 773 2614 4 93 v 833 2614 V 773 2617 64 4 v 146 2798 a(The) p 350 2798 a(follo) m(wing) p 765 2798 a(lemma) p 1083 2798 a(is) p 1184 2798 a(useful) p 1467 2798 a(to) p 1589 2798 a(determine) p 2048 2798 a(the) p 2219 2798 a(de\014ciency) p 2669 2798 a(indices) p 2993 2798 a(of) p Fn 3107 2798 a(L) p Fm 3173 2813 a(N) p Fo 3241 2798 a(.) p 3321 2798 a(In) p 3446 2798 a(the) p 3617 2798 a(sequel,) 0 2954 y(w) m(e) p 145 2954 a(denote) p Fm 490 2879 a(N) p Fk 483 2954 a(\010) p Fm 461 3020 a(j) p Fl 494 3020 a(=1) p Fk 600 2954 a(H) p Fm 684 2969 a(j) p Fo 754 2954 a(the) p 924 2954 a(orthogonal) p 1413 2954 a(direct) p 1691 2954 a(sum) p 1898 2954 a(of) p 2011 2954 a(Hilb) s(ert) p 2348 2954 a(spaces) p Fk 2649 2954 a(H) p Fl 2733 2969 a(1) p Fn 2772 2954 a(;) p 2816 2954 a(:) p 2860 2954 a(:) p 2904 2954 a(:) p 2947 2954 a(;) p Fk 2991 2954 a(H) p Fm 3075 2969 a(N) p Fo 3176 2954 a(\() p Fn(N) p Fo 3332 2954 a(=) p 3438 2954 a(1) p Fn(;) p Fo 3531 2954 a(2) p Fn(;) p 3624 2954 a(:) p 3668 2954 a(:) p 3712 2954 a(:) p 3754 2954 a(;) p Fo 3815 2954 a(or) p Fn 0 3178 a(N) p Fo 116 3178 a(=) p Fk 220 3178 a(1) p Fo(\).) p 428 3178 a(The) p 628 3178 a(norm) p 883 3178 a(of) p Fm 1024 3103 a(N) p Fk 1017 3178 a(\010) p Fm 994 3244 a(j) p Fl 1027 3244 a(=1) p Fk 1133 3178 a(H) p Fm 1217 3193 a(j) p Fo 1286 3178 a(is) p 1384 3178 a(de\014ned) p 1720 3178 a(b) m(y) p Fk 1265 3518 a(jj) p Fo(\() p Fn(u) p Fl 1415 3533 a(1) p Fn 1454 3518 a(;) p 1498 3518 a(:) p 1542 3518 a(:) p 1586 3518 a(:) p 1628 3518 a(;) p 1672 3518 a(u) p Fm 1728 3533 a(N) p Fo 1795 3518 a(\)) p Fk(jj) p Fl 1889 3477 a(2) p Fg 1889 3543 a(\010H) p Fe 2004 3553 a(j) p Fo 2067 3518 a(=) p Fm 2201 3410 a(N) p Fi 2172 3435 a(X) p Fm 2171 3617 a(j) p Fl 2204 3617 a(=1) p Fk 2310 3518 a(jj) p Fn(u) p Fm 2422 3533 a(j) p Fk 2457 3518 a(jj) p Fl 2513 3477 a(2) p Fg 2513 3543 a(H) p Fe 2573 3553 a(j) p Fn 2610 3518 a(;) p Fo 0 3811 a(where) p Fn 277 3811 a(u) p Fm 333 3826 a(j) p Fk 396 3811 a(2) p 491 3811 a(H) p Fm 575 3826 a(j) p Fo 639 3811 a(for) p Fn 783 3811 a(j) p Fo 856 3811 a(=) p 960 3811 a(1) p Fn(;) p Fk 1053 3811 a(\001) p 1098 3811 a(\001) p 1143 3811 a(\001) p Fn 1185 3811 a(;) p 1229 3811 a(N) p Fo 1317 3811 a(.) p 1386 3811 a(In) p 1503 3811 a(particular,) p 1976 3811 a(when) p Fk 2225 3811 a(fH) p Fm 2359 3826 a(j) p Fk 2396 3811 a(g) p Fm 2446 3775 a(N) 2446 3836 y(j) p Fl 2479 3836 a(=1) p Fo 2600 3811 a(are) p 2757 3811 a(m) m(utually) p 3164 3811 a(orthogonal) p 3647 3811 a(closed) 0 3985 y(subspaces) p 446 3985 a(of) p 558 3985 a(a) p 639 3985 a(Hilb) s(ert) p 976 3985 a(space) p Fk 1237 3985 a(H) p Fo 1322 3985 a(,) p 1382 3985 a(w) m(e) p 1526 3985 a(iden) m(tify) p Fm 1911 3911 a(N) p Fk 1904 3985 a(\010) p Fm 1881 4052 a(j) p Fl 1914 4052 a(=1) p Fk 2020 3985 a(H) p Fm 2104 4000 a(j) p Fo 2174 3985 a(with) p 2396 3985 a(the) p 2565 3985 a(closed) p 2853 3985 a(subspace) p 3261 3985 a(of) p Fk 3372 3985 a(H) p Fo 3490 3985 a(generated) 0 4163 y(b) m(y) p Fk 135 4163 a(fH) p Fm 269 4178 a(j) p Fk 306 4163 a(g) p Fm 356 4126 a(N) 356 4187 y(j) p Fl 389 4187 a(=1) p Fo 482 4163 a(.) 0 4283 y(Remark.) p 405 4283 a(When) p Fn 690 4283 a(N) p Fo 806 4283 a(=) p Fk 910 4283 a(1) p Fo(,) p 1070 4283 a(the) p 1238 4283 a(note) p 1455 4283 a(`) p Fn(j) p Fo 1556 4283 a(=) p 1659 4283 a(1) p Fn(;) p Fk 1752 4283 a(\001) p 1797 4283 a(\001) p 1842 4283 a(\001) p Fn 1885 4283 a(;) p 1929 4283 a(N) p Fo 2017 4283 a(') p 2077 4283 a(ab) s(o) m(v) m(e) p 2353 4283 a(implies) p 2684 4283 a(`for) p 2861 4283 a(ev) m(ery) p 3118 4283 a(p) s(ositiv) m(e) p 3479 4283 a(in) m(teger) p Fn 3802 4283 a(j) p Fo 3848 4283 a('.) 0 4403 y(W) p 92 4403 a(e) p 164 4403 a(often) p 407 4403 a(use) p 571 4403 a(the) p 736 4403 a(notation) p 1122 4403 a(of) p 1229 4403 a(this) p 1415 4403 a(t) m(yp) s(e) p 1631 4403 a(in) p 1741 4403 a(the) p 1905 4403 a(sequel,) p 2220 4403 a(to) p 2335 4403 a(simplify) p 2700 4403 a(the) p 2864 4403 a(statemen) m(ts;) p 3380 4403 a(for) p 3525 4403 a(example,) 0 4524 y(the) p 168 4524 a(note) p 385 4524 a(`) p Fn(u) p Fl 468 4539 a(1) p Fo 529 4524 a(+) p Fk 627 4524 a(\001) p 672 4524 a(\001) p 717 4524 a(\001) p Fo 765 4524 a(+) p Fn 863 4524 a(u) p Fm 919 4539 a(N) p Fo 986 4524 a(') p 1046 4524 a(implies) p 1377 4524 a(`) p Fi 1404 4457 a(P) p Fg 1492 4484 a(1) p Fm 1492 4548 a(j) p Fl 1525 4548 a(=1) p Fn 1635 4524 a(u) p Fm 1691 4539 a(j) p Fo 1727 4524 a(',) p 1814 4524 a(when) p Fn 2068 4524 a(N) p Fo 2185 4524 a(=) p Fk 2288 4524 a(1) p Fo(.) p Fq 146 4723 a(Lemma) p 543 4723 a(4.4) p Ff 735 4723 a(L) p 791 4723 a(et) p Fo 908 4723 a(1) p Fk 992 4723 a(\024) p Fn 1104 4723 a(N) p 1228 4723 a(<) p Fk 1339 4723 a(1) p Ff 1477 4723 a(or) p Fn 1607 4723 a(N) p Fo 1731 4723 a(=) p Fk 1842 4723 a(1) p Ff(.) p 2028 4723 a(L) p 2084 4723 a(et) p Fn 2200 4723 a(\037) p Fk 2296 4723 a(2) p Fn 2398 4723 a(C) p Fg 2475 4687 a(1) p Fl 2468 4747 a(0) p Fo 2550 4723 a(\() p Fk(fj) p Fn(z) p Fk 2715 4723 a(j) p Fn 2778 4723 a(<) p Fm 2899 4684 a(R) p 2899 4700 54 4 v Fl 2908 4757 a(3) p Fk 2962 4723 a(g) p Fo(\)) p Ff 3089 4723 a(with) p Fo 3304 4723 a(0) p Fk 3388 4723 a(\024) p Fn 3501 4723 a(\037) p Fk 3597 4723 a(\024) p Fo 3710 4723 a(1) p Ff 3797 4723 a(on) p Fj 0 4843 a(R) p Fl 87 4801 a(2) p Ff 127 4843 a(,) p Fn 199 4843 a(\037) p Fo 299 4843 a(=) p 413 4843 a(1) p Ff 503 4843 a(on) p Fk 648 4843 a(fj) p Fn(z) p Fk 775 4843 a(j) p Fn 841 4843 a(<) p Fm 966 4804 a(R) p 966 4820 V Fl 975 4878 a(4) p Fk 1029 4843 a(g) p Ff 1120 4843 a(and) p 1315 4843 a(put) p Fn 1490 4843 a(\037) p Fm 1551 4858 a(j) p Fo 1588 4843 a(\() p Fn(z) p Fo 1675 4843 a(\)) p 1752 4843 a(=) p Fn 1867 4843 a(\037) p Fo(\() p Fn(z) p Fk 2042 4843 a(\000) p Fn 2146 4843 a(z) p Fm 2191 4858 a(j) p Fo 2228 4843 a(\)) p Ff(.) p 2358 4843 a(L) p 2414 4843 a(et) p Fn 2532 4843 a(t) p Fg 2567 4858 a(\000) p Fm(z) p Fe 2655 4868 a(j) p Ff 2732 4843 a(b) p 2772 4843 a(e) p 2858 4843 a(the) p 3026 4843 a(magnetic) p 3447 4843 a(tr) p 3515 4843 a(anslation) 0 4964 y(fr) p 66 4964 a(om) p Fk 230 4964 a(fj) p Fn(z) p Fk 380 4964 a(\000) p Fn 479 4964 a(z) p Fm 524 4979 a(j) p Fk 561 4964 a(j) p Fn 617 4964 a(<) p Fm 730 4924 a(R) p 730 4940 V Fl 739 4998 a(2) p Fk 794 4964 a(g) p Ff 878 4964 a(to) p Fk 995 4964 a(fj) p Fn(z) p Fk 1122 4964 a(j) p Fn 1178 4964 a(<) p Fm 1291 4924 a(R) p 1291 4940 V Fl 1300 4998 a(2) p Fk 1355 4964 a(g) p Ff 1440 4964 a(intertwining) p Fk 1989 4964 a(L) p Fm 2058 4979 a(N) p Ff 2160 4964 a(with) p Fk 2371 4964 a(L) p Fm 2440 4909 a(\013) p Fe 2485 4919 a(j) p Fl 2440 4985 a(1) p Ff 2522 4964 a(.) p 2597 4964 a(Then,) p 2881 4964 a(the) p 3043 4964 a(fol) p 3153 4964 a(lowing) p 3456 4964 a(holds.) p Fo 1902 5214 a(22) p eop %%Page: 23 23 23 22 bop Ff 146 631 a(\(i\)) p 291 631 a(The) p 490 631 a(map) p Fn 1066 751 a(T) p Fm 1123 766 a(j) p Fo 1187 751 a(:) p Fn 1242 751 a(D) p Fo 1326 751 a(\() p Fn(L) p Fg 1430 710 a(\003) p Fm 1430 776 a(N) p Fo 1497 751 a(\)) p Fk 1563 751 a(3) p Fn 1657 751 a(u) p Fk 1740 751 a(7!) p Fn 1868 751 a(t) p Fg 1903 766 a(\000) p Fm(z) p Fe 1991 776 a(j) p Fo 2027 751 a(\() p Fn(\037) p Fm 2126 766 a(j) p Fn 2163 751 a(u) p Fo(\)) p Fk 2284 751 a(2) p Fn 2378 751 a(D) p Fo 2462 751 a(\(\() p Fn(L) p Fm 2604 697 a(\013) p Fe 2649 707 a(j) p Fl 2604 773 a(1) p Fo 2686 751 a(\)) p Fg 2724 710 a(\003) p Fo 2763 751 a(\)) p 3653 751 a(\(4.10\)) p Ff 0 926 a(is) p 105 926 a(wel) p 245 926 a(l-de\014ne) p 550 926 a(d) p 633 926 a(and) p 822 926 a(c) p 862 926 a(ontinuous) p 1313 926 a(for) p 1468 926 a(e) p 1508 926 a(ach) p Fn 1687 926 a(j) p Fo 1761 926 a(=) p 1865 926 a(1) p Fn(;) p 1958 926 a(:) p 2002 926 a(:) p 2046 926 a(:) p 2088 926 a(;) p 2132 926 a(N) p Ff 2220 926 a(.) p 2295 926 a(Mor) p 2468 926 a(e) p 2508 926 a(over,) p 2754 926 a(the) p 2916 926 a(map) p Fn 906 1181 a(T) p Fo 1005 1181 a(:) p Fn 1060 1181 a(D) p Fo 1144 1181 a(\() p Fn(L) p Fg 1248 1140 a(\003) p Fm 1248 1205 a(N) p Fo 1315 1181 a(\)) p Fk 1380 1181 a(3) p Fn 1475 1181 a(u) p Fk 1558 1181 a(7!) p Fo 1685 1181 a(\() p Fn(T) p Fl 1780 1196 a(1) p Fn 1820 1181 a(u;) p 1920 1181 a(:) p 1964 1181 a(:) p 2008 1181 a(:) p 2050 1181 a(;) p 2094 1181 a(T) p Fm 2151 1196 a(N) p Fn 2218 1181 a(u) p Fo(\)) p Fk 2339 1181 a(2) p Fm 2463 1106 a(N) p Fk 2456 1181 a(\010) p Fm 2433 1247 a(j) p Fl 2466 1247 a(=1) p Fn 2572 1181 a(D) p Fo 2656 1181 a(\(\() p Fn(L) p Fm 2798 1127 a(\013) p Fe 2843 1137 a(j) p Fl 2798 1203 a(1) p Fo 2880 1181 a(\)) p Fg 2918 1140 a(\003) p Fo 2958 1181 a(\)) p 3653 1181 a(\(4.11\)) p Ff 0 1458 a(is) p 104 1458 a(wel) p 244 1458 a(l-de\014ne) p 549 1458 a(d) p 631 1458 a(and) p 819 1458 a(c) p 859 1458 a(ontinuous,) p 1338 1458 a(wher) p 1534 1458 a(e) p Fk 1613 1458 a(\010) p Fm 1690 1422 a(N) 1690 1483 y(j) p Fl 1723 1483 a(=1) p Fn 1833 1458 a(D) p Fo 1917 1458 a(\(\() p Fn(L) p Fm 2059 1404 a(\013) p Fe 2104 1414 a(j) p Fl 2059 1480 a(1) p Fo 2141 1458 a(\)) p Fg 2179 1422 a(\003) p Fo 2219 1458 a(\)) p Ff 2290 1458 a(is) p 2394 1458 a(the) p 2555 1458 a(ortho) p 2773 1458 a(gonal) p 3031 1458 a(dir) p 3147 1458 a(e) p 3187 1458 a(ct) p 3297 1458 a(sum) p 3503 1458 a(of) p 3617 1458 a(Hilb) p 3784 1458 a(ert) 0 1579 y(sp) p 85 1579 a(ac) p 175 1579 a(es) p Fk 294 1579 a(f) p Fn(D) p Fo 428 1579 a(\(\() p Fn(L) p Fm 570 1525 a(\013) p Fe 615 1535 a(j) p Fl 570 1601 a(1) p Fo 652 1579 a(\)) p Fg 690 1543 a(\003) p Fo 729 1579 a(\)) p Fk(g) p Fm 817 1543 a(N) 817 1603 y(j) p Fl 850 1603 a(=1) p Ff 943 1579 a(.) 146 1724 y(\(ii\)) p 321 1724 a(We) p 498 1724 a(have) p Fn 722 1724 a(T) p Fm 779 1739 a(j) p Fn 816 1724 a(D) p Fo 900 1724 a(\() p 938 1646 134 4 v Fn(L) p Fm 1004 1739 a(N) p Fo 1071 1724 a(\)) p Fk 1136 1724 a(\032) p Fn 1242 1724 a(D) p Fo 1326 1724 a(\() p 1364 1631 149 4 v Fn(L) p Fm 1430 1670 a(\013) p Fe 1475 1680 a(j) p Fl 1430 1746 a(1) p Fo 1512 1724 a(\)) p Ff(,) p 1614 1724 a(for) p 1770 1724 a(every) p Fn 2028 1724 a(j) p Fo 2101 1724 a(=) p 2205 1724 a(1) p Fn(;) p 2298 1724 a(:) p 2342 1724 a(:) p 2386 1724 a(:) p 2428 1724 a(;) p 2472 1724 a(N) p Ff 2560 1724 a(.) 146 1844 y(\(iii\)) p 351 1844 a(The) p 550 1844 a(map) p Fn 328 2105 a(S) p Fo 421 2105 a(:) p Fm 506 2030 a(N) p Fk 498 2105 a(\010) p Fm 476 2171 a(j) p Fl 509 2171 a(=1) p Fn 615 2105 a(D) p Fo 699 2105 a(\(\() p Fn(L) p Fm 841 2051 a(\013) p Fe 886 2061 a(j) p Fl 841 2127 a(1) p Fo 923 2105 a(\)) p Fg 961 2064 a(\003) p Fo 1000 2105 a(\)) p Fk 1066 2105 a(3) p Fo 1160 2105 a(\() p Fn(u) p Fl 1254 2120 a(1) p Fn 1293 2105 a(;) p 1337 2105 a(:) p 1381 2105 a(:) p 1425 2105 a(:) p 1468 2105 a(;) p 1512 2105 a(u) p Fm 1568 2120 a(N) p Fo 1634 2105 a(\)) p Fk 1700 2105 a(7!) p Fn 1827 2105 a(t) p Fm 1862 2120 a(z) p Fd 1895 2129 a(1) p Fo 1934 2105 a(\() p Fn(\037u) p Fl 2089 2120 a(1) p Fo 2128 2105 a(\)) p 2188 2105 a(+) p Fn 2286 2105 a(:) p 2330 2105 a(:) p 2374 2105 a(:) p Fo 2423 2105 a(+) p Fn 2521 2105 a(t) p Fm 2556 2120 a(z) p Fe 2589 2131 a(N) p Fo 2650 2105 a(\() p Fn(\037u) p Fm 2805 2120 a(N) p Fo 2872 2105 a(\)) p Fk 2938 2105 a(2) p Fn 3032 2105 a(D) p Fo 3116 2105 a(\() p Fn(L) p Fg 3220 2064 a(\003) p Fm 3220 2130 a(N) p Fo 3287 2105 a(\)) p 3653 2105 a(\(4.12\)) p Ff 0 2369 a(is) p 105 2369 a(wel) p 245 2369 a(l-de\014ne) p 550 2369 a(d) p 633 2369 a(and) p 822 2369 a(c) p 862 2369 a(ontinuous.) 146 2490 y(\(iv\)) p 336 2490 a(We) p 513 2490 a(have) p Fn 737 2490 a(S) p Fo 803 2490 a(\() p Fk(\010) p Fm 918 2453 a(N) 918 2514 y(j) p Fl 951 2514 a(=1) p Fn 1061 2490 a(D) p Fo 1145 2490 a(\() p 1183 2397 V Fn(L) p Fm 1249 2435 a(\013) p Fe 1294 2445 a(j) p Fl 1249 2511 a(1) p Fo 1331 2490 a(\)\)) p Fk 1435 2490 a(\032) p Fn 1540 2490 a(D) p Fo 1624 2490 a(\() p 1662 2411 134 4 v Fn(L) p Fm 1728 2505 a(N) p Fo 1795 2490 a(\)) p Ff(.) 146 2620 y(\(v\)) p 306 2620 a(A) n(n) p 465 2620 a(element) p Fn 826 2620 a(u) p Fk 910 2620 a(2) p Fn 1004 2620 a(D) p Fo 1088 2620 a(\() p Fn(L) p Fg 1192 2584 a(\003) p Fm 1192 2645 a(N) p Fo 1259 2620 a(\)) p Ff 1332 2620 a(b) p 1372 2620 a(elongs) p 1666 2620 a(to) p Fn 1783 2620 a(D) p Fo 1867 2620 a(\() p 1905 2542 V Fn(L) p Fm 1971 2635 a(N) p Fo 2038 2620 a(\)) p Ff 2111 2620 a(if) p 2205 2620 a(and) p 2395 2620 a(only) p 2606 2620 a(if) p Fn 1163 2852 a(T) p Fm 1220 2867 a(j) p Fn 1257 2852 a(u) p Fk 1340 2852 a(2) p Fn 1434 2852 a(D) p Fo 1518 2852 a(\() p 1556 2759 149 4 v Fn(L) p Fm 1622 2798 a(\013) p Fe 1667 2808 a(j) p Fl 1622 2874 a(1) p Fo 1704 2852 a(\)) p 1777 2852 a(for) p 1926 2852 a(ev) m(ery) p Fn 2184 2852 a(j) p Fo 2258 2852 a(=) p 2362 2852 a(1) p Fn(;) p 2455 2852 a(:) p 2499 2852 a(:) p 2543 2852 a(:) p 2585 2852 a(;) p 2629 2852 a(N) p 2712 2852 a(:) p Fo 3653 2852 a(\(4.13\)) p Ff 146 3071 a(\(vi\)) p 336 3071 a(F) p 393 3071 a(or) p 518 3071 a(every) p Fo 775 3071 a(\() p Fn(u) p Fl 869 3086 a(1) p Fn 908 3071 a(;) p 952 3071 a(:) p 996 3071 a(:) p 1040 3071 a(:) p 1083 3071 a(;) p 1127 3071 a(u) p Fm 1183 3086 a(N) p Fo 1249 3071 a(\)) p Fk 1315 3071 a(2) p 1409 3071 a(\010) p Fm 1487 3035 a(N) 1487 3096 y(j) p Fl 1520 3096 a(=1) p Fn 1630 3071 a(D) p Fo 1714 3071 a(\(\() p Fn(L) p Fm 1856 3017 a(\013) p Fe 1901 3027 a(j) p Fl 1856 3093 a(1) p Fo 1938 3071 a(\)) p Fg 1976 3035 a(\003) p Fo 2015 3071 a(\)) p Ff(,) p 2118 3071 a(we) p 2262 3071 a(have) p Fo 1179 3345 a(\() p Fn(T) p 1288 3345 a(S) p Fk 1376 3345 a(\000) p Fn 1475 3345 a(I) p 1526 3345 a(d) p Fo(\)\() p Fn(u) p Fl 1709 3360 a(1) p Fn 1748 3345 a(;) p 1792 3345 a(:) p 1836 3345 a(:) p 1880 3345 a(:) p 1923 3345 a(;) p 1967 3345 a(u) p Fm 2023 3360 a(N) p Fo 2089 3345 a(\)) p Fk 2155 3345 a(2) p Fm 2278 3270 a(N) p Fk 2271 3345 a(\010) p Fm 2249 3411 a(j) p Fl 2282 3411 a(=1) p Fn 2388 3345 a(D) p Fo 2472 3345 a(\() p 2510 3252 V Fn(L) p Fm 2576 3291 a(\013) p Fe 2621 3301 a(j) p Fl 2576 3367 a(1) p Fo 2658 3345 a(\)) p Fn(:) p Ff 146 3623 a(Pr) p 248 3623 a(o) p 293 3623 a(of.) p Fo 447 3623 a(\(i\)) p 593 3623 a(Let) p Fn 779 3623 a(u) p Fk 882 3623 a(2) p Fn 995 3623 a(D) p Fo 1079 3623 a(\() p Fn(L) p Fg 1183 3586 a(\003) p Fm 1183 3647 a(N) p Fo 1250 3623 a(\).) p 1392 3623 a(Then) p Fn 1658 3623 a(\037) p Fm 1719 3638 a(j) p Fn 1756 3623 a(u) p Fk 1858 3623 a(2) p Fn 1972 3623 a(D) p Fo 2056 3623 a(\() p Fn(L) p Fg 2160 3586 a(\003) p Fm 2160 3647 a(N) p Fo 2227 3623 a(\)) p 2309 3623 a(b) m(y) p 2455 3623 a(Lemma) p 2815 3623 a(4.3.) p 3044 3623 a(By) p 3208 3623 a(the) p 3387 3623 a(in) m(tert) m(wining) 0 3743 y(prop) s(ert) m(y) p 398 3743 a(and) p 588 3743 a(unitarit) m(y) p 994 3743 a(of) p Fn 1106 3743 a(t) p Fg 1141 3758 a(\000) p Fm(z) p Fe 1229 3768 a(j) p Fo 1265 3743 a(,) p 1325 3743 a(w) m(e) p 1468 3743 a(ha) m(v) m(e) p Fk 707 3963 a(jj) p Fn(\037) p Fm 824 3978 a(j) p Fn 860 3963 a(u) p Fk(jj) p Fl 972 3922 a(2) p Fm 972 3987 a(N) p Fo 1121 3963 a(=) p Fk 1280 3963 a(jjL) p Fm 1405 3978 a(N) p Fo 1471 3963 a(\() p Fn(\037) p Fm 1570 3978 a(j) p Fn 1607 3963 a(u) p Fo(\)) p Fk(jj) p Fl 1757 3922 a(2) p Fo 1817 3963 a(+) p Fk 1915 3963 a(jj) p Fn(\037) p Fm 2032 3978 a(j) p Fn 2068 3963 a(u) p Fk(jj) p Fl 2180 3922 a(2) p Fo 1121 4108 a(=) p Fk 1280 4108 a(jjL) p Fm 1405 4054 a(\013) p Fe 1450 4064 a(j) p Fl 1405 4130 a(1) p Fn 1486 4108 a(t) p Fg 1521 4123 a(\000) p Fm(z) p Fe 1609 4133 a(j) p Fn 1646 4108 a(\037) p Fm 1707 4123 a(j) p Fn 1743 4108 a(u) p Fk(jj) p Fl 1855 4067 a(2) p Fo 1916 4108 a(+) p Fk 2014 4108 a(jj) p Fn(t) p Fg 2105 4123 a(\000) p Fm(z) p Fe 2193 4133 a(j) p Fn 2229 4108 a(\037) p Fm 2290 4123 a(j) p Fn 2326 4108 a(u) p Fk(jj) p Fl 2438 4067 a(2) p Fo 2504 4108 a(=) p Fk 2608 4108 a(jj) p Fn(t) p Fg 2699 4123 a(\000) p Fm(z) p Fe 2787 4133 a(j) p Fn 2823 4108 a(\037) p Fm 2884 4123 a(j) p Fn 2920 4108 a(u) p Fk(jj) p Fl 3032 4067 a(2) 3032 4133 y(1) p Fm(;\013) p Fe 3132 4143 a(j) p Fn 3168 4108 a(;) p Fo 3653 4108 a(\(4.14\)) 0 4343 y(where) p Fk 281 4343 a(jj) p Fn(v) p Fk 388 4343 a(jj) p Fl 444 4306 a(2) 444 4367 y(1) p Fm(;\013) p Fe 544 4377 a(j) p Fo 607 4343 a(=) p Fk 711 4343 a(jjL) p Fm 836 4289 a(\013) p Fe 881 4299 a(j) p Fl 836 4364 a(1) p Fn 917 4343 a(v) p Fk 968 4343 a(jj) p Fl 1024 4306 a(2) p Fo 1083 4343 a(+) p Fk 1179 4343 a(jj) p Fn(v) p Fk 1286 4343 a(jj) p Fl 1342 4306 a(2) p Fo 1412 4343 a(for) p Fn 1560 4343 a(v) p Fk 1638 4343 a(2) p Fn 1732 4343 a(D) p Fo 1816 4343 a(\(\() p Fn(L) p Fm 1958 4289 a(\013) p Fe 2003 4299 a(j) p Fl 1958 4364 a(1) p Fo 2040 4343 a(\)) p Fg 2078 4306 a(\003) p Fo 2118 4343 a(\).) p 2226 4343 a(Th) m(us) p Fn 2472 4343 a(t) p Fg 2507 4358 a(\000) p Fm(z) p Fe 2595 4368 a(j) p Fn 2632 4343 a(\037) p Fm 2693 4358 a(j) p Fn 2729 4343 a(u) p Fk 2813 4343 a(2) p Fn 2907 4343 a(D) p Fo 2991 4343 a(\(\() p Fn(L) p Fm 3133 4289 a(\013) p Fe 3178 4299 a(j) p Fl 3133 4364 a(1) p Fo 3215 4343 a(\)) p Fg 3253 4306 a(\003) p Fo 3292 4343 a(\)) p 3362 4343 a(and) p 3551 4343 a(the) p 3718 4343 a(map) p Fn 0 4463 a(T) p Fm 57 4478 a(j) p Fo 125 4463 a(is) p 222 4463 a(w) m(ell-de\014ned.) p 794 4463 a(The) p 994 4463 a(con) m(tin) m(uit) m(y) p 1453 4463 a(of) p Fn 1563 4463 a(T) p Fm 1620 4478 a(j) p Fo 1689 4463 a(follo) m(ws) p 2008 4463 a(from) p 2238 4463 a(\(4.14\)) p 2518 4463 a(and) p 2707 4463 a(\(4.6\).) p 2978 4463 a(Moreo) m(v) m(er,) p 3432 4463 a(w) m(e) p 3575 4463 a(ha) m(v) m(e) p 3799 4463 a(b) m(y) 0 4583 y(\(4.14\)) p 282 4583 a(and) p 471 4583 a(\(4.6\)) p Fm 557 4770 a(N) p Fi 529 4794 a(X) p Fm 527 4977 a(j) p Fl 560 4977 a(=1) p Fk 667 4877 a(jj) p Fn(T) p Fm 780 4892 a(j) p Fn 816 4877 a(u) p Fk(jj) p Fl 928 4836 a(2) 928 4902 y(1) p Fk 994 4877 a(\024) p Fn 1099 4877 a(C) p Fl 1169 4892 a(1) p Fm 1254 4770 a(N) p Fi 1226 4794 a(X) p Fm 1225 4977 a(j) p Fl 1258 4977 a(=1) p Fi 1364 4760 a(Z) p Fg 1410 4949 a(fj) p Fm(z) p Fg 1501 4949 a(\000) p Fm(z) p Fe 1589 4959 a(j) p Fg 1621 4949 a(j) p Fm(<) p Fe 1706 4922 a(R) p 1705 4934 47 4 v Fd 1713 4975 a(2) p Fg 1761 4949 a(g) p Fo 1801 4877 a(\() p Fk(jL) p Fm 1936 4892 a(N) p Fn 2002 4877 a(u) p Fk(j) p Fl 2086 4836 a(2) p Fo 2147 4877 a(+) p Fk 2245 4877 a(j) p Fn(u) p Fk(j) p Fl 2357 4836 a(2) p Fo 2395 4877 a(\)) p Fn(dxdy) p Fk 2669 4877 a(\024) p Fn 2774 4877 a(C) p Fl 2844 4892 a(1) p Fk 2883 4877 a(jj) p Fn(u) p Fk(jj) p Fl 3051 4836 a(2) p Fm 3051 4902 a(N) p Fn 3144 4877 a(<) p Fk 3248 4877 a(1) p Fn(;) p Fo 3653 4877 a(\(4.15\)) 1902 5214 y(23) p eop %%Page: 24 24 24 23 bop Fo 0 650 a(where) p Fn 278 650 a(C) p Fl 348 665 a(1) p Fo 415 650 a(is) p 509 650 a(a) p 587 650 a(constan) m(t) p 976 650 a(dep) s(enden) m(t) p 1444 650 a(only) p 1654 650 a(on) p Fn 1785 650 a(\037) p Fo(.) p 1915 650 a(W) p 2007 650 a(e) p 2079 650 a(used) p 2298 650 a(that) p 2505 650 a(the) p 2669 650 a(balls) p Fi 2893 554 a(n) p Fk 2948 650 a(fj) p Fn(z) p Fk 3098 650 a(\000) p Fn 3197 650 a(z) p Fm 3242 665 a(j) p Fk 3279 650 a(j) p Fn 3335 650 a(<) p Fm 3448 611 a(R) p 3448 627 54 4 v Fl 3457 684 a(2) p Fk 3512 650 a(g) p Fi 3562 554 a(o) p Fm 3617 577 a(N) 3617 704 y(j) p Fl 3650 704 a(=1) p Fo 3772 650 a(are) 0 800 y(disjoin) m(t) p 347 800 a(b) m(y) p 483 800 a(the) p 651 800 a(assumption) p 1167 800 a(\(1.1\).) p 1438 800 a(Th) m(us) p 1685 800 a(the) p 1853 800 a(map) p Fn 2070 800 a(T) p Fo 2173 800 a(is) p 2271 800 a(w) m(ell-de\014ned) p 2805 800 a(and) p 2995 800 a(con) m(tin) m(uous.) 146 920 y(\(ii\)) p 314 920 a(Since) p Fn 574 920 a(T) p Fm 631 935 a(j) p Fo 705 920 a(maps) p Fn 965 920 a(D) p Fo 1049 920 a(\() p Fn(L) p Fm 1153 935 a(N) p Fo 1220 920 a(\)) p 1294 920 a(=) p Fn 1406 920 a(C) p Fg 1483 884 a(1) p Fl 1476 945 a(0) p Fo 1558 920 a(\() p Fj(R) p Fl 1683 878 a(2) p Fk 1748 920 a(n) p Fn 1823 920 a(S) p Fm 1883 935 a(N) p Fo 1951 920 a(\)) p 2026 920 a(in) m(to) p Fn 2229 920 a(D) p Fo 2313 920 a(\() p Fn(L) p Fm 2417 866 a(\013) p Fe 2462 876 a(j) p Fl 2417 942 a(1) p Fo 2499 920 a(\)) p 2573 920 a(=) p Fn 2685 920 a(C) p Fg 2762 884 a(1) p Fl 2755 945 a(0) p Fo 2836 920 a(\() p Fj(R) p Fl 2962 878 a(2) p Fk 3027 920 a(n) p 3102 920 a(f) p Fo(0) p Fk(g) p Fo(\),) p 3354 920 a(the) p 3527 920 a(assertion) 0 1041 y(holds) p 255 1041 a(from) p 486 1041 a(the) p 654 1041 a(con) m(tin) m(uit) m(y) p 1114 1041 a(of) p 1225 1041 a(the) p 1393 1041 a(map) p Fn 1610 1041 a(T) p Fm 1667 1056 a(j) p Fo 1704 1041 a(.) 146 1161 y(\(iii\)) p 334 1161 a(T) p 396 1161 a(ak) m(e) p 568 1161 a(\() p Fn(u) p Fl 662 1176 a(1) p Fn 701 1161 a(;) p 745 1161 a(:) p 789 1161 a(:) p 833 1161 a(:) p 875 1161 a(;) p 919 1161 a(u) p Fm 975 1176 a(N) p Fo 1042 1161 a(\)) p Fk 1108 1161 a(2) p 1202 1161 a(\010) p Fm 1279 1125 a(N) 1279 1186 y(j) p Fl 1312 1186 a(=1) p Fn 1422 1161 a(D) p Fo 1506 1161 a(\(\() p Fn(L) p Fm 1648 1107 a(\013) p Fe 1693 1117 a(j) p Fl 1648 1183 a(1) p Fo 1730 1161 a(\)) p Fg 1768 1125 a(\003) p Fo 1808 1161 a(\).) p 1915 1161 a(By) p 2067 1161 a(the) p 2233 1161 a(unitarit) m(y) p 2637 1161 a(and) p 2825 1161 a(the) p 2991 1161 a(in) m(tert) m(wining) p 3536 1161 a(prop) s(ert) m(y) 0 1292 y(of) p Fn 111 1292 a(t) p Fm 146 1307 a(z) p Fe 179 1317 a(j) p Fo 216 1292 a(,) p 275 1292 a(w) m(e) p 419 1292 a(ha) m(v) m(e) p Fn 644 1292 a(t) p Fm 679 1307 a(z) p Fe 712 1317 a(j) p Fo 749 1292 a(\() p Fn(\037u) p Fm 904 1307 a(j) p Fo 940 1292 a(\)) p Fk 1005 1292 a(2) p Fn 1099 1292 a(L) p Fl 1165 1256 a(2) p Fo 1205 1292 a(\() p Fj(R) p Fl 1331 1250 a(2) p Fo 1370 1292 a(\),) p Fk 1468 1292 a(L) p Fm 1537 1307 a(N) p Fn 1603 1292 a(t) p Fm 1638 1307 a(z) p Fe 1671 1317 a(j) p Fo 1708 1292 a(\() p Fn(\037u) p Fm 1863 1307 a(j) p Fo 1899 1292 a(\)) p 1965 1292 a(=) p Fn 2068 1292 a(t) p Fm 2103 1307 a(z) p Fe 2136 1317 a(j) p Fk 2173 1292 a(L) p Fl 2242 1307 a(1) p Fo 2281 1292 a(\() p Fn(\037u) p Fm 2436 1307 a(j) p Fo 2472 1292 a(\)) p Fk 2538 1292 a(2) p Fn 2632 1292 a(L) p Fl 2698 1256 a(2) p Fo 2738 1292 a(\() p Fj(R) p Fl 2863 1250 a(2) p Fo 2903 1292 a(\)) p 2973 1292 a(and) p Fk 1416 1501 a(jj) p Fn(t) p Fm 1507 1516 a(z) p Fe 1540 1526 a(j) p Fo 1576 1501 a(\() p Fn(\037u) p Fm 1731 1516 a(j) p Fo 1767 1501 a(\)) p Fk(jj) p Fl 1861 1460 a(2) p Fm 1861 1526 a(N) p Fo 1955 1501 a(=) p Fk 2059 1501 a(jj) p Fn(\037u) p Fm 2232 1516 a(j) p Fk 2267 1501 a(jj) p Fl 2323 1460 a(2) 2323 1526 y(1) p Fm(;\013) p Fe 2423 1536 a(j) p Fn 2459 1501 a(:) p Fo 3653 1501 a(\(4.16\)) 0 1710 y(Th) m(us) p 248 1710 a(w) m(e) p 393 1710 a(ha) m(v) m(e) p Fn 620 1710 a(t) p Fm 655 1725 a(z) p Fe 688 1735 a(j) p Fo 724 1710 a(\() p Fn(\037u) p Fm 879 1725 a(j) p Fo 916 1710 a(\)) p Fk 984 1710 a(2) p Fn 1080 1710 a(D) p Fo 1164 1710 a(\() p Fn(L) p Fg 1268 1674 a(\003) p Fm 1268 1735 a(N) p Fo 1335 1710 a(\)) p 1407 1710 a(b) m(y) p 1544 1710 a(\(i\)) p 1681 1710 a(of) p 1793 1710 a(Prop) s(osition) p 2319 1710 a(4.2.) p 2519 1710 a(When) p Fn 2805 1710 a(N) p Fo 2927 1710 a(is) p 3026 1710 a(\014nite,) p 3304 1710 a(the) p 3474 1710 a(con) m(tin) m(uit) m(y) 0 1830 y(of) p 111 1830 a(the) p 279 1830 a(map) p 496 1830 a(follo) m(ws) p 816 1830 a(from) p 1046 1830 a(\(4.16\).) 146 1966 y(Consider) p 542 1966 a(the) p 700 1966 a(case) p Fn 895 1966 a(N) p Fo 1011 1966 a(=) p Fk 1115 1966 a(1) p Fo(.) p 1281 1966 a(Since) p Fi 1525 1869 a(n) p Fo 1581 1966 a(supp) q(\() p Fn(t) p Fm 1855 1981 a(z) p Fe 1888 1991 a(j) p Fo 1924 1966 a(\() p Fn(\037u) p Fm 2079 1981 a(j) p Fo 2116 1966 a(\)\)) p Fi 2192 1869 a(o) p Fg 2247 1892 a(1) p Fm 2247 2020 a(j) p Fl 2280 2020 a(=1) p Fo 2395 1966 a(are) p 2547 1966 a(disjoin) m(t,) p 2913 1966 a(w) m(e) p 3045 1966 a(ha) m(v) m(e) p 3260 1966 a(that) p Fk 3460 1966 a(f) p Fn(t) p Fm 3545 1981 a(z) p Fe 3578 1991 a(j) p Fo 3615 1966 a(\() p Fn(\037u) p Fm 3770 1981 a(j) p Fo 3806 1966 a(\)) p Fk(g) p Fg 3894 1929 a(1) p Fm 3894 1990 a(j) p Fl 3927 1990 a(=1) p Fo 0 2115 a(is) p 98 2115 a(a) p 179 2115 a(m) m(utually) p 591 2115 a(orthogonal) p 1079 2115 a(sequence) p 1483 2115 a(in) p Fn 1597 2115 a(D) p Fo 1681 2115 a(\() p Fn(L) p Fg 1785 2079 a(\003) p Fm 1785 2140 a(N) p Fo 1852 2115 a(\).) p 1961 2115 a(Th) m(us) p 2208 2115 a(w) m(e) p 2351 2115 a(ha) m(v) m(e) p 2576 2115 a(b) m(y) p 2712 2115 a(\(4.16\)) p 2994 2115 a(and) p 3183 2115 a(\(4.6\)) p Fk 1292 2401 a(jj) p Fm 1407 2291 a(N) p Fd 1463 2300 a(2) p Fi 1393 2318 a(X) p Fm 1364 2502 a(j) p Fl 1397 2502 a(=) p Fm(N) p Fd 1508 2511 a(1) p Fn 1558 2401 a(t) p Fm 1593 2416 a(z) p Fe 1626 2426 a(j) p Fo 1663 2401 a(\() p Fn(\037u) p Fm 1818 2416 a(j) p Fo 1854 2401 a(\)) p Fk(jj) p Fl 1948 2360 a(2) p Fm 1948 2426 a(N) p Fo 2042 2401 a(=) p Fm 2189 2291 a(N) p Fd 2245 2300 a(2) p Fi 2175 2318 a(X) p Fm 2146 2502 a(j) p Fl 2179 2502 a(=) p Fm(N) p Fd 2290 2511 a(1) p Fk 2340 2401 a(jj) p Fn(t) p Fm 2431 2416 a(z) p Fe 2464 2426 a(j) p Fo 2500 2401 a(\() p Fn(\037u) p Fm 2655 2416 a(j) p Fo 2691 2401 a(\)) p Fk(jj) p Fl 2785 2360 a(2) p Fm 2785 2426 a(N) p Fo 1133 2720 a(=) p Fm 1336 2610 a(N) p Fd 1392 2619 a(2) p Fi 1321 2637 a(X) p Fm 1292 2820 a(j) p Fl 1325 2820 a(=) p Fm(N) p Fd 1436 2829 a(1) p Fk 1486 2720 a(jj) p Fn(\037u) p Fm 1659 2735 a(j) p Fk 1695 2720 a(jj) p Fl 1751 2678 a(2) 1751 2744 y(1) p Fm(;\013) p Fe 1851 2754 a(j) p Fk 1915 2720 a(\024) p Fn 2020 2720 a(C) p Fl 2090 2735 a(1) p Fm 2189 2610 a(N) p Fd 2245 2619 a(2) p Fi 2175 2637 a(X) p Fm 2146 2820 a(j) p Fl 2179 2820 a(=) p Fm(N) p Fd 2290 2829 a(1) p Fk 2340 2720 a(jj) p Fn(u) p Fm 2452 2735 a(j) p Fk 2488 2720 a(jj) p Fl 2544 2678 a(2) 2544 2744 y(1) p Fm(;\013) p Fe 2644 2754 a(j) p Fo 3653 2720 a(\(4.17\)) 0 3008 y(for) p 150 3008 a(1) p Fk 229 3008 a(\024) p Fn 336 3008 a(N) p Fl 414 3023 a(1) p Fn 483 3008 a(<) p 589 3008 a(N) p Fl 667 3023 a(2) p Fn 736 3008 a(<) p Fk 842 3008 a(1) p Fo(,) p 1002 3008 a(where) p Fn 1285 3008 a(C) p Fl 1355 3023 a(1) p Fo 1428 3008 a(is) p 1528 3008 a(a) p 1610 3008 a(constan) m(t) p 2005 3008 a(dep) s(enden) m(t) p 2477 3008 a(only) p 2693 3008 a(on) p Fn 2829 3008 a(\037) p Fo(.) p 2965 3008 a(The) p 3166 3008 a(last) p 3352 3008 a(expression) p 3823 3008 a(of) 0 3128 y(\(4.17\)) p 287 3128 a(con) m(v) m(erges) p 729 3128 a(to) p 854 3128 a(0) p 941 3128 a(as) p Fn 1066 3128 a(N) p Fl 1144 3143 a(1) p Fo 1221 3128 a(and) p Fn 1416 3128 a(N) p Fl 1494 3143 a(2) p Fo 1572 3128 a(go) s(es) p 1792 3128 a(to) p 1916 3128 a(in\014nit) m(y) p 2212 3128 a(,) p 2278 3128 a(since) p 2522 3128 a(\() p Fn(u) p Fm 2616 3143 a(j) p Fo 2652 3128 a(\)) p Fg 2690 3092 a(1) p Fm 2690 3153 a(j) p Fl 2723 3153 a(=1) p Fk 2854 3128 a(2) p 2957 3128 a(\010) p Fg 3034 3092 a(1) p Fm 3034 3153 a(j) p Fl 3067 3153 a(=1) p Fn 3178 3128 a(D) p Fo 3262 3128 a(\(\() p Fn(L) p Fm 3404 3074 a(\013) p Fe 3449 3084 a(j) p Fl 3404 3150 a(1) p Fo 3486 3128 a(\)) p Fg 3524 3092 a(\003) p Fo 3563 3128 a(\).) p 3687 3128 a(Th) m(us) 0 3248 y(the) p 172 3248 a(con) m(v) m(ergence) p 714 3248 a(of) p 829 3248 a(the) p 1001 3248 a(sum) p 1211 3248 a(follo) m(ws) p 1535 3248 a(from) p 1768 3248 a(the) p 1940 3248 a(Cauc) m(h) m(y) p 2293 3248 a(criterion.) p 2732 3248 a(Putting) p Fn 3095 3248 a(N) p Fl 3173 3263 a(1) p Fo 3246 3248 a(=) p 3356 3248 a(1) p 3440 3248 a(and) p 3634 3248 a(taking) 0 3369 y(the) p 168 3369 a(limit) p Fn 401 3369 a(N) p Fl 479 3384 a(2) p Fk 546 3369 a(!) p 674 3369 a(1) p Fo 806 3369 a(in) p 920 3369 a(\(4.17\),) p 1228 3369 a(w) m(e) p 1372 3369 a(obtain) p 1676 3369 a(the) p 1844 3369 a(con) m(tin) m(uit) m(y) p 2304 3369 a(statemen) m(t.) 146 3489 y(\(iv\)) p 333 3489 a(The) p 534 3489 a(set) p Fn 877 3698 a(D) p Fl 958 3713 a(0) p Fo 1080 3698 a(=) p Fk 1239 3698 a(f) p Fo(\() p Fn(u) p Fl 1383 3713 a(1) p Fn 1422 3698 a(;) p 1466 3698 a(:) p 1510 3698 a(:) p 1554 3698 a(:) p 1597 3698 a(;) p 1641 3698 a(u) p Fm 1697 3713 a(n) p Fn 1743 3698 a(;) p Fo 1787 3698 a(0) p Fn(;) p Fo 1880 3698 a(0) p Fn(;) p 1973 3698 a(:) p 2017 3698 a(:) p 2061 3698 a(:) p 2103 3698 a(;) p Fo 2147 3698 a(0\);) p Fn 2309 3698 a(n) p Fk 2395 3698 a(\024) p Fn 2500 3698 a(N) p 2583 3698 a(;) p 2660 3698 a(n) p 2746 3698 a(<) p Fk 2849 3698 a(1) p Fn(;) 1239 3844 y(u) p Fl 1295 3859 a(1) p Fn 1334 3844 a(;) p 1378 3844 a(:) p 1422 3844 a(:) p 1466 3844 a(:) p 1509 3844 a(;) p 1553 3844 a(u) p Fm 1609 3859 a(n) p Fk 1683 3844 a(2) p Fn 1777 3844 a(C) p Fg 1854 3803 a(1) p Fl 1847 3868 a(0) p Fo 1929 3844 a(\() p Fj(R) p Fl 2054 3802 a(2) p Fk 2116 3844 a(n) p 2187 3844 a(f) p Fo(0) p Fk(g) p Fo(\)) p Fk(g) p Fo 0 4065 a(is) p 94 4065 a(dense) p 356 4065 a(in) p Fk 465 4065 a(\010) p Fm 543 4029 a(N) 543 4090 y(j) p Fl 576 4090 a(=1) p Fn 686 4065 a(D) p Fo 770 4065 a(\() p 808 3972 149 4 v Fn(L) p Fm 874 4011 a(\013) p Fe 919 4021 a(j) p Fl 874 4087 a(1) p Fo 956 4065 a(\)) p 1022 4065 a(and) p Fn 1208 4065 a(S) p 1274 4065 a(D) p Fl 1355 4080 a(0) p Fk 1421 4065 a(\032) p Fn 1527 4065 a(C) p Fg 1604 4029 a(1) p Fl 1597 4090 a(0) p Fo 1678 4065 a(\() p Fj(R) p Fl 1804 4023 a(2) p Fk 1856 4065 a(n) p Fn 1919 4065 a(S) p Fm 1979 4080 a(N) p Fo 2047 4065 a(\)) p 2113 4065 a(=) p Fn 2216 4065 a(D) p Fo 2300 4065 a(\() p Fn(L) p Fm 2404 4080 a(N) p Fo 2471 4065 a(\).) p 2578 4065 a(Th) m(us) p 2821 4065 a(the) p 2985 4065 a(assertion) p 3388 4065 a(follo) m(ws) p 3704 4065 a(from) 0 4185 y(the) p 168 4185 a(con) m(tin) m(uit) m(y) p 629 4185 a(of) p Fn 740 4185 a(S) p Fo 806 4185 a(.) 146 4306 y(\(v\)) p 320 4306 a(By) p 486 4306 a(\(ii\),) p 693 4306 a(it) p 804 4306 a(is) p 916 4306 a(su\016cien) m(t) p 1339 4306 a(to) p 1472 4306 a(sho) m(w) p 1727 4306 a(that) p Fn 1952 4306 a(u) p Fk 2058 4306 a(2) p Fn 2176 4306 a(D) p Fo 2260 4306 a(\() p Fn(L) p Fg 2364 4270 a(\003) p Fm 2364 4330 a(N) p Fo 2431 4306 a(\)) p 2515 4306 a(and) p 2718 4306 a(\(4.13\)) p 3014 4306 a(implies) p Fn 3358 4306 a(u) p Fk 3465 4306 a(2) p Fn 3582 4306 a(D) p Fo 3666 4306 a(\() p 3704 4228 134 4 v Fn(L) p Fm 3770 4321 a(N) p Fo 3837 4306 a(\).) 0 4426 y(Supp) s(ose) p 383 4426 a(an) p 518 4426 a(elemen) m(t) p Fn 878 4426 a(u) p Fo 967 4426 a(in) p Fn 1080 4426 a(D) p Fo 1164 4426 a(\() p Fn(L) p Fg 1268 4390 a(\003) p Fm 1268 4451 a(N) p Fo 1336 4426 a(\)) p 1406 4426 a(satis\014es) p 1765 4426 a(\(4.13\).) p 2085 4426 a(By) p 2238 4426 a(\(iv\),) p 2452 4426 a(w) m(e) p 2596 4426 a(ha) m(v) m(e) p Fn 1222 4635 a(S) p 1288 4635 a(T) p 1359 4635 a(u) p Fo 1442 4635 a(=) p Fn 1545 4635 a(S) p Fo 1611 4635 a(\() p Fn(T) p Fl 1706 4650 a(1) p Fn 1746 4635 a(u;) p 1846 4635 a(:) p 1890 4635 a(:) p 1934 4635 a(:) p 1976 4635 a(;) p 2020 4635 a(T) p Fm 2077 4650 a(N) p Fn 2144 4635 a(u) p Fo(\)) p Fk 2265 4635 a(2) p Fn 2359 4635 a(D) p Fo 2443 4635 a(\() p 2481 4557 V Fn(L) p Fm 2547 4650 a(N) p Fo 2615 4635 a(\)) p Fn(:) p Fo 0 4844 a(Th) m(us) p 247 4844 a(it) p 345 4844 a(is) p 443 4844 a(su\016cien) m(t) p 852 4844 a(to) p 972 4844 a(sho) m(w) p Fn 1454 4965 a(v) p Fo 1533 4965 a(=) p Fn 1636 4965 a(u) p Fk 1714 4965 a(\000) p Fn 1814 4965 a(S) p 1880 4965 a(T) p 1951 4965 a(u) p Fk 2033 4965 a(2) p Fn 2127 4965 a(D) p Fo 2211 4965 a(\() p 2249 4887 V Fn(L) p Fm 2315 4980 a(N) p Fo 2383 4965 a(\)) p Fn(:) p Fo 3653 4965 a(\(4.18\)) 1902 5214 y(24) p eop %%Page: 25 25 25 24 bop Fo 0 631 a(By) p 153 631 a(\(i\)) p 289 631 a(and) p 478 631 a(\(iii\),) p 695 631 a(w) m(e) p 839 631 a(ha) m(v) m(e) p Fn 1064 631 a(v) p Fk 1142 631 a(2) p Fn 1236 631 a(D) p Fo 1320 631 a(\() p Fn(L) p Fg 1424 595 a(\003) p Fm 1424 656 a(N) p Fo 1491 631 a(\).) p 1600 631 a(Moreo) m(v) m(er,) p 2055 631 a(w) m(e) p 2198 631 a(ha) m(v) m(e) 1485 894 y(supp) p Fn 1702 894 a(v) p Fk 1781 894 a(\032) p Fj 1886 894 a(R) p Fl 1973 852 a(2) p Fk 2035 894 a(n) p Fn 2107 894 a(U) p Fi 2200 773 a(\022) p Fn 2271 827 a(R) p 2271 871 76 4 v Fo 2284 963 a(4) p Fi 2356 773 a(\023) p Fo 0 1146 a(b) m(y) p 135 1146 a(de\014nition) p 569 1146 a(of) p Fn 680 1146 a(T) p Fm 737 1161 a(j) p Fo 806 1146 a(and) p Fn 996 1146 a(S) p Fo 1062 1146 a(.) p 1132 1146 a(Th) m(us) p 1379 1146 a(\(4.18\)) p 1661 1146 a(follo) m(ws) p 1982 1146 a(from) p 2212 1146 a(\(ii\)) p 2374 1146 a(of) p 2486 1146 a(Prop) s(osition) p 3010 1146 a(4.2.) 146 1267 y(\(vi\)) p 333 1267 a(By) p 486 1267 a(de\014nition,) 638 1530 y(supp) p Fk 839 1530 a(f) p Fo(\() p Fn(T) p Fm 984 1545 a(j) p Fn 1021 1530 a(S) p Fo 1087 1530 a(\)\() p Fn(u) p Fl 1219 1545 a(1) p Fn 1257 1530 a(;) p 1301 1530 a(:) p 1345 1530 a(:) p 1389 1530 a(:) p 1432 1530 a(;) p 1476 1530 a(u) p Fm 1532 1545 a(N) p Fo 1599 1530 a(\)) p Fk 1659 1530 a(\000) p Fn 1758 1530 a(u) p Fm 1814 1545 a(j) p Fk 1850 1530 a(g) p Fo 1928 1530 a(=) p 2031 1530 a(supp) q(\() p Fn(\037) p Fl 2331 1489 a(2) p Fk 2393 1530 a(\000) p Fo 2493 1530 a(1\)) p Fn(u) p Fm 2636 1545 a(j) p Fk 2699 1530 a(\032) p 2804 1530 a(fj) p Fn(z) p Fk 2931 1530 a(j) p 2987 1530 a(\025) p Fn 3102 1463 a(R) p 3102 1507 V Fo 3115 1598 a(4) p Fk 3187 1530 a(g) p Fn(:) p Fo 0 1776 a(Th) m(us) p 247 1776 a(the) p 415 1776 a(conclusion) p 887 1776 a(follo) m(ws) p 1207 1776 a(from) p 1438 1776 a(\(ii\)) p 1600 1776 a(of) p 1711 1776 a(Prop) s(osition) p 2236 1776 a(4.2.) p 2431 1708 64 4 v 2431 1801 4 93 v 2491 1801 V 2431 1804 64 4 v 146 1998 a(W) p 238 1998 a(e) p 314 1998 a(shall) p 542 1998 a(determine) p 998 1998 a(the) p 1166 1998 a(de\014ciency) p 1613 1998 a(indices) p 1933 1998 a(of) p Fn 2044 1998 a(L) p Fm 2110 2013 a(N) p Fo 2178 1998 a(.) p Fq 146 2197 a(Prop) s(osition) p 752 2197 a(4.5) p Ff 944 2197 a(L) p 1000 2197 a(et) p Fo 1116 2197 a(1) p Fk 1201 2197 a(\024) p Fn 1314 2197 a(N) p 1438 2197 a(<) p Fk 1549 2197 a(1) p Ff 1688 2197 a(or) p Fn 1818 2197 a(N) p Fo 1942 2197 a(=) p Fk 2054 2197 a(1) p Ff(.) p 2241 2197 a(Then,) p 2530 2197 a(the) p 2696 2197 a(de\014ciency) p 3152 2197 a(indic) p 3357 2197 a(es) p 3480 2197 a(of) p Fn 3598 2197 a(L) p Fm 3664 2212 a(N) p Ff 3771 2197 a(ar) p 3857 2197 a(e) p Fo 0 2318 a(\(2) p Fn(N) p 170 2318 a(;) p Fo 214 2318 a(2) p Fn(N) p Fo 351 2318 a(\)) p Ff(.) 146 2540 y(Pr) p 248 2540 a(o) p 293 2540 a(of.) p Fo 446 2540 a(Since) p 712 2540 a(the) p 891 2540 a(op) s(erator) p Fn 1294 2540 a(L) p Fm 1360 2555 a(N) p Fo 1471 2540 a(is) p 1580 2540 a(symmetric) p 2066 2540 a(and) p 2266 2540 a(p) s(ositiv) m(e,) p 2668 2540 a(the) p 2847 2540 a(de\014ciency) p 3305 2540 a(indices) p Fn 3636 2540 a(m) p Fg 3721 2555 a(\006) p Fo 3826 2540 a(=) 0 2661 y(dim) p 179 2661 a(Ker\() p Fn(L) p Fg 440 2624 a(\003) p Fm 440 2685 a(N) p Fk 520 2661 a(\007) p Fn 609 2661 a(i) p Fo(\)) p 709 2661 a(are) p 867 2661 a(equal) p 1120 2661 a(\(see) p 1310 2661 a([Re-Si,) p 1622 2661 a(Corollary) p 2048 2661 a(of) p 2155 2661 a(Theorem) p 2562 2661 a(X.1]\).) p 2845 2661 a(Since) p Fn 3095 2661 a(D) p Fo 3179 2661 a(\() p Fn(L) p Fg 3283 2624 a(\003) p Fm 3283 2685 a(N) p Fo 3350 2661 a(\)) p 3415 2661 a(=) p Fn 3519 2661 a(D) p Fo 3603 2661 a(\() p 3641 2583 134 4 v Fn(L) p Fm 3707 2676 a(N) p Fo 3774 2661 a(\)) p Fk 3824 2661 a(\010) p Fo 0 2781 a(Ker\() p Fn(L) p Fg 261 2745 a(\003) p Fm 261 2806 a(N) p Fk 352 2781 a(\000) p Fn 453 2781 a(i) p Fo(\)) p Fk 547 2781 a(\010) p Fo 648 2781 a(Ker\() p Fn(L) p Fg 909 2745 a(\003) p Fm 909 2806 a(N) p Fo 999 2781 a(+) p Fn 1098 2781 a(i) p Fo(\)) p 1204 2781 a(\(see) p 1401 2781 a([Re-Si,) p 1718 2781 a(\(b\)) p 1883 2781 a(of) p 1995 2781 a(Lemma) p 2345 2781 a(in) p 2460 2781 a(page) p 2689 2781 a(138]\),) p 2962 2781 a(it) p 3061 2781 a(is) p 3161 2781 a(su\016cien) m(t) p 3572 2781 a(to) p 3693 2781 a(sho) m(w) 0 2901 y(that) 1399 3022 y(dim) p Fn 1578 3022 a(D) p Fo 1662 3022 a(\() p Fn(L) p Fg 1766 2981 a(\003) p Fm 1766 3046 a(N) p Fo 1833 3022 a(\)) p Fn(=D) p Fo 2004 3022 a(\() p 2042 2944 V Fn(L) p Fm 2108 3037 a(N) p Fo 2175 3022 a(\)) p 2241 3022 a(=) p 2344 3022 a(4) p Fn(N) p 2476 3022 a(;) p Fo 3653 3022 a(\(4.19\)) 0 3194 y(where) p Fn 282 3194 a(D) p Fo 366 3194 a(\() p Fn(L) p Fg 470 3158 a(\003) p Fm 470 3219 a(N) p Fo 537 3194 a(\)) p Fn(=D) p Fo 708 3194 a(\() p 746 3116 V Fn(L) p Fm 812 3209 a(N) p Fo 879 3194 a(\)) p 949 3194 a(is) p 1048 3194 a(the) p 1216 3194 a(quotien) m(t) p 1600 3194 a(Banac) m(h) p 1949 3194 a(space) p 2209 3194 a(equipp) s(ed) p 2626 3194 a(with) p 2849 3194 a(the) p 3017 3194 a(norm) p Fk 1160 3409 a(jj) p Fo([) p Fn(u) p Fo(]) p Fk(jj) p Fl 1382 3368 a(2) p Fm 1382 3433 a(N) p Fo 1475 3409 a(=) p 1578 3409 a(inf) p Fk 1697 3409 a(fjj) p Fn(u) p Fo 1879 3409 a(+) p Fn 1977 3409 a(v) p Fk 2028 3409 a(jj) p Fl 2084 3368 a(2) p Fm 2084 3433 a(N) p Fo 2151 3409 a(;) p Fn 2227 3409 a(v) p Fk 2305 3409 a(2) p Fn 2399 3409 a(D) p Fo 2483 3409 a(\() p 2521 3331 V Fn(L) p Fm 2587 3424 a(N) p Fo 2655 3409 a(\)) p Fk(g) p Fo 0 3624 a(\([) p Fn(u) p Fo(]) p 180 3624 a(is) p 278 3624 a(the) p 446 3624 a(equiv) p 668 3624 a(alence) p 961 3624 a(class) p 1190 3624 a(of) p Fn 1301 3624 a(u) p Fo(\).) p 1465 3624 a(Since) p Fn 1720 3624 a(D) p Fo 1804 3624 a(\() p Fn(L) p Fg 1908 3588 a(\003) p Fm 1908 3648 a(N) p Fo 1975 3624 a(\)) p 2045 3624 a(is) p 2143 3624 a(a) p 2225 3624 a(Hilb) s(ert) p 2561 3624 a(space,) p 2848 3624 a(w) m(e) p 2992 3624 a(ha) m(v) m(e) p Fk 1560 3839 a(jj) p Fo([) p Fn(u) p Fo(]) p Fk(jj) p Fm 1782 3854 a(N) p Fo 1875 3839 a(=) p Fk 1979 3839 a(jj) p Fn(P) p Fm 2098 3854 a(N) p Fn 2164 3839 a(u) p Fk(jj) p Fm 2276 3854 a(N) p Fo 0 4054 a(for) p Fn 148 4054 a(u) p Fk 231 4054 a(2) p Fn 325 4054 a(D) p Fo 409 4054 a(\() p Fn(L) p Fg 513 4017 a(\003) p Fm 513 4078 a(N) p Fo 580 4054 a(\),) p 677 4054 a(where) p Fn 958 4054 a(P) p Fm 1021 4069 a(N) p Fo 1119 4054 a(denotes) p 1471 4054 a(the) p 1637 4054 a(orthogonal) p 2124 4054 a(pro) p 2270 4054 a(jection) p 2586 4054 a(on) m(to) p Fn 2805 4054 a(D) p Fo 2889 4054 a(\() p 2927 3975 V Fn(L) p Fm 2993 4069 a(N) p Fo 3060 4054 a(\)) p Fg 3098 4017 a(?) p Fo 3184 4054 a(=) p 3288 4054 a(Ker\() p Fn(L) p Fg 3549 4017 a(\003) p Fm 3549 4078 a(N) p Fk 3636 4054 a(\000) p Fn 3734 4054 a(i) p Fo(\)) p Fk 3825 4054 a(\010) p Fo 0 4174 a(Ker\() p Fn(L) p Fg 261 4138 a(\003) p Fm 261 4199 a(N) p Fo 351 4174 a(+) p Fn 449 4174 a(i) p Fo(\).) 146 4306 y(De\014ne) p 448 4306 a(a) p 530 4306 a(map) p Fi 767 4274 a(e) p Fn 747 4306 a(T) p Fo 850 4306 a(from) p Fn 1080 4306 a(D) p Fo 1164 4306 a(\() p Fn(L) p Fg 1268 4270 a(\003) p Fm 1268 4331 a(N) p Fo 1335 4306 a(\)) p Fn(=D) p Fo 1506 4306 a(\() p 1544 4228 V Fn(L) p Fm 1610 4321 a(N) p Fo 1677 4306 a(\)) p 1748 4306 a(to) p Fk 1867 4306 a(\010) p Fm 1944 4270 a(N) 1944 4331 y(j) p Fl 1977 4331 a(=1) p Fn 2088 4306 a(D) p Fo 2172 4306 a(\(\() p Fn(L) p Fm 2314 4252 a(\013) p Fe 2359 4262 a(j) p Fl 2314 4328 a(1) p Fo 2396 4306 a(\)) p Fg 2434 4270 a(\003) p Fo 2473 4306 a(\)) p Fn(=D) p Fo 2644 4306 a(\() p 2682 4214 149 4 v Fn(L) p Fm 2748 4252 a(\013) p Fe 2793 4262 a(j) p Fl 2748 4328 a(1) p Fo 2830 4306 a(\)) p 2900 4306 a(b) m(y) p Fi 1615 4543 a(e) p Fn 1594 4575 a(T) p Fo 1665 4575 a(\([) p Fn(u) p Fo(]\)) p 1878 4575 a(=) p Fm 2012 4500 a(N) p Fk 2004 4575 a(\010) p Fm 1982 4641 a(j) p Fl 2015 4641 a(=1) p Fo 2104 4575 a([) p Fn(T) p Fm 2188 4590 a(j) p Fn 2225 4575 a(u) p Fo(]) 0 4840 y(for) p Fn 157 4840 a(u) p Fk 253 4840 a(2) p Fn 360 4840 a(D) p Fo 444 4840 a(\() p Fn(L) p Fg 548 4804 a(\003) p Fm 548 4865 a(N) p Fo 615 4840 a(\),) p 723 4840 a(where) p Fn 1012 4840 a(T) p Fm 1069 4855 a(j) p Fo 1146 4840 a(is) p 1252 4840 a(the) p 1428 4840 a(op) s(erator) p 1828 4840 a(de\014ned) p 2172 4840 a(in) p 2293 4840 a(Lemma) p 2649 4840 a(4.4.) p 2868 4840 a(W) p 2960 4840 a(e) p 3043 4840 a(see) p 3209 4840 a(that) p 3428 4840 a(the) p 3603 4840 a(v) m(ectors) p Fk 0 4960 a(f) p Fo([) p Fn(T) p Fm 134 4975 a(j) p Fn 171 4960 a(u) p Fo(]) p Fk(g) p Fm 304 4924 a(N) 304 4985 y(j) p Fl 337 4985 a(=1) p Fo 465 4960 a(are) p 630 4960 a(indep) s(enden) m(t) p 1185 4960 a(of) p 1299 4960 a(the) p 1469 4960 a(c) m(hoice) p 1761 4960 a(of) p 1875 4960 a(a) p 1959 4960 a(represen) m(tativ) m(e) p Fn 2588 4960 a(u) p Fo 2678 4960 a(in) p 2794 4960 a(a) p 2878 4960 a(equiv) p 3100 4960 a(alence) p 3395 4960 a(class) p 3627 4960 a([) p Fn(u) p Fo(],) p 3799 4960 a(b) m(y) 1902 5214 y(25) p eop %%Page: 26 26 26 25 bop Fo 0 631 a(\(ii\)) p 163 631 a(of) p 275 631 a(Lemma) p 623 631 a(4.4.) p 820 631 a(Since) p Fn 1075 631 a(u) p Fk 1153 631 a(\000) p Fn 1253 631 a(P) p Fm 1316 646 a(N) p Fn 1383 631 a(u) p Fk 1467 631 a(2) p Fn 1562 631 a(D) p Fo 1646 631 a(\() p 1684 553 134 4 v Fn(L) p Fm 1750 646 a(N) p Fo 1817 631 a(\),) p 1915 631 a(w) m(e) p 2059 631 a(ha) m(v) m(e) p 2285 631 a([) p Fn(u) p Fo(]) p 2423 631 a(=) p 2527 631 a([) p Fn(P) p Fm 2617 646 a(N) p Fn 2684 631 a(u) p Fo(]) p 2800 631 a(and) p 2990 631 a(th) m(us) p 3205 631 a([) p Fn(T) p Fm 3289 646 a(j) p Fn 3326 631 a(u) p Fo(]) p 3437 631 a(=) p 3541 631 a([) p Fn(T) p Fm 3625 646 a(j) p Fn 3662 631 a(P) p Fm 3725 646 a(N) p Fn 3792 631 a(u) p Fo(].) 0 751 y(Th) m(us) p 247 751 a(w) m(e) p 391 751 a(ha) m(v) m(e) p 616 751 a(b) m(y) p 751 751 a(\(4.15\)) p Fm 680 938 a(N) p Fi 651 963 a(X) p Fm 650 1145 a(j) p Fl 683 1145 a(=1) p Fk 789 1046 a(jj) p Fo([) p Fn(T) p Fm 929 1061 a(j) p Fn 965 1046 a(u) p Fo(]) p Fk(jj) p Fl 1104 1005 a(2) 1104 1070 y(1) p Fm(;\013) p Fe 1204 1080 a(j) p Fo 1268 1046 a(=) p Fm 1401 938 a(N) p Fi 1372 963 a(X) p Fm 1371 1145 a(j) p Fl 1404 1145 a(=1) p Fk 1510 1046 a(jj) p Fo([) p Fn(T) p Fm 1650 1061 a(j) p Fn 1686 1046 a(P) p Fm 1749 1061 a(N) p Fn 1817 1046 a(u) p Fo(]) p Fk(jj) p Fl 1956 1005 a(2) 1956 1070 y(1) p Fm(;\013) p Fe 2056 1080 a(j) p Fk 2119 1046 a(\024) p Fn 2224 1046 a(C) p Fl 2294 1061 a(1) p Fk 2333 1046 a(jj) p Fn(P) p Fm 2452 1061 a(N) p Fn 2519 1046 a(u) p Fk(jj) p Fl 2631 1005 a(2) p Fm 2631 1070 a(N) p Fo 2725 1046 a(=) p Fn 2828 1046 a(C) p Fl 2898 1061 a(1) p Fk 2937 1046 a(jj) p Fo([) p Fn(u) p Fo(]) p Fk(jj) p Fl 3159 1005 a(2) p Fm 3159 1070 a(N) p Fn 3225 1046 a(:) p Fo 0 1366 a(Th) m(us) p Fi 279 1334 a(e) p Fn 259 1366 a(T) p Fo 374 1366 a(is) p 484 1366 a(w) m(ell-de\014ned) p 1030 1366 a(and) p 1232 1366 a(con) m(tin) m(uous.) p 1794 1366 a(Moreo) m(v) m(er,) p 2264 1366 a(w) m(e) p 2419 1366 a(ha) m(v) m(e) p 2656 1366 a(that) p Fi 2900 1334 a(e) p Fn 2880 1366 a(T) p Fo 2995 1366 a(is) p 3105 1366 a(injectiv) m(e) p 3504 1366 a(b) m(y) p 3652 1366 a(\(v\)) p 3823 1366 a(of) 0 1487 y(Lemma) p 348 1487 a(4.4.) 146 1607 y(W) p 238 1607 a(e) p 309 1607 a(shall) p 532 1607 a(sho) m(w) p 768 1607 a(that) p Fi 995 1575 a(e) p Fn 974 1607 a(T) p Fo 1072 1607 a(is) p 1165 1607 a(surjectiv) m(e.) p 1638 1607 a(T) p 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a(1) p Fo 1127 1752 a(=) p Fk 1231 1752 a(jj) p Fn(P) p Fm 1364 1698 a(\013) p Fe 1409 1708 a(j) p Fl 1350 1774 a(1) p Fn 1445 1752 a(u) p Fm 1501 1767 a(j) p Fk 1537 1752 a(jj) p Fl 1593 1767 a(1) p Fo 1664 1752 a(\() p Fn(P) p Fm 1779 1698 a(\013) p Fe 1824 1708 a(j) p Fl 1765 1774 a(1) p Fo 1892 1752 a(is) p 1990 1752 a(the) p 2157 1752 a(orthogonal) p 2645 1752 a(pro) p 2791 1752 a(jection) p 3108 1752 a(on) m(to) p Fn 3327 1752 a(D) p Fo 3411 1752 a(\() p 3449 1660 V Fn(L) p Fm 3515 1698 a(\013) p Fe 3560 1708 a(j) p Fl 3515 1774 a(1) p Fo 3597 1752 a(\)) p Fg 3635 1716 a(?) p Fo 3694 1752 a(\),) p 3791 1752 a(w) m(e) 0 1888 y(ma) m(y) p 211 1888 a(assume) p Fn 548 1888 a(u) p Fm 604 1903 a(j) p Fk 668 1888 a(2) p Fn 762 1888 a(D) p Fo 846 1888 a(\() p 884 1795 V Fn(L) p Fm 950 1834 a(\013) p Fe 995 1844 a(j) p Fl 950 1910 a(1) p Fo 1032 1888 a(\)) p Fg 1070 1852 a(?) p Fo 1162 1888 a(and) p Fk 1351 1888 a(jj) p Fo([) p Fn(u) p Fm 1490 1903 a(j) p Fo 1526 1888 a(]) p Fk(jj) p Fl 1609 1903 a(1) p Fo 1675 1888 a(=) p Fk 1779 1888 a(jj) p Fn(u) p Fm 1891 1903 a(j) p Fk 1926 1888 a(jj) p Fl 1982 1903 a(1) p Fo 2021 1888 a(.) p 2092 1888 a(Then) p 2346 1888 a(w) m(e) p 2490 1888 a(ha) m(v) m(e) p Fm 1294 2077 a(N) p Fi 1265 2102 a(X) p Fm 1264 2285 a(j) p Fl 1297 2285 a(=1) p Fk 1403 2185 a(jj) p Fn(u) p Fm 1515 2200 a(j) p Fk 1551 2185 a(jj) p Fl 1607 2144 a(2) 1607 2210 y(1) p Fm(;\013) p Fe 1707 2220 a(j) p Fo 1771 2185 a(=) p Fm 1904 2077 a(N) p Fi 1875 2102 a(X) p Fm 1874 2285 a(j) p Fl 1907 2285 a(=1) p Fk 2013 2185 a(jj) p Fo([) p Fn(u) p Fm 2152 2200 a(j) p Fo 2188 2185 a(]) p Fk(jj) p Fl 2271 2144 a(2) 2271 2210 y(1) p Fm(;\013) p Fe 2371 2220 a(j) p Fn 2435 2185 a(<) p Fk 2538 2185 a(1) p Fo 0 2496 a(and) p 194 2496 a(th) m(us) p 413 2496 a(\() p Fn(u) p Fl 507 2511 a(1) p Fn 546 2496 a(;) p 590 2496 a(:) p 634 2496 a(:) p 678 2496 a(:) p 720 2496 a(;) p 764 2496 a(u) p Fm 820 2511 a(N) p Fo 887 2496 a(\)) p Fk 960 2496 a(2) p 1061 2496 a(\010) p Fm 1138 2460 a(N) 1138 2521 y(j) p Fl 1171 2521 a(=1) p Fn 1282 2496 a(D) p Fo 1366 2496 a(\(\() p Fn(L) p Fm 1508 2442 a(\013) p Fe 1553 2452 a(j) p Fl 1508 2518 a(1) p Fo 1590 2496 a(\)) p Fg 1628 2460 a(\003) p Fo 1667 2496 a(\).) p 1788 2496 a(By) p 1945 2496 a(\(vi\)) p 2137 2496 a(of) p 2252 2496 a(Lemma) p 2604 2496 a(4.4,) p 2794 2496 a(w) m(e) p 2942 2496 a(ha) m(v) m(e) p 3171 2496 a(\([) p Fn(u) p Fl 3292 2511 a(1) p Fo 3331 2496 a(]) p Fn(;) p 3402 2496 a(:) p 3446 2496 a(:) p 3490 2496 a(:) p 3533 2496 a(;) p Fo 3577 2496 a([) p Fn(u) p Fm 3660 2511 a(N) p Fo 3726 2496 a(]\)) p 3826 2496 a(=) p Fi 20 2602 a(e) p Fn 0 2635 a(T) p 71 2635 a(S) p Fo 137 2635 a(\() p Fn(u) p Fl 231 2650 a(1) p Fn 269 2635 a(;) p 313 2635 a(:) p 357 2635 a(:) p 401 2635 a(:) p 444 2635 a(;) p 488 2635 a(u) p Fm 544 2650 a(N) p Fo 611 2635 a(\)) p Fk 676 2635 a(2) p Fo 770 2635 a(Ran) p Fi 982 2602 a(e) p Fn 962 2635 a(T) p Fo 1032 2635 a(.) 146 2755 y(Therefore) p 595 2755 a(w) m(e) p 745 2755 a(ha) m(v) m(e) p 977 2755 a(the) p 1152 2755 a(map) p Fi 1396 2723 a(e) p Fn 1375 2755 a(T) p Fo 1486 2755 a(is) p 1590 2755 a(a) p 1679 2755 a(bijectiv) m(e) p 2073 2755 a(con) m(tin) m(uous) p 2568 2755 a(map) p 2792 2755 a(from) p 3029 2755 a(a) p 3117 2755 a(Hilb) s(ert) p 3460 2755 a(space) p 3727 2755 a(to) p 3853 2755 a(a) 0 2875 y(Hilb) s(ert) p 347 2875 a(space.) p 679 2875 a(This) p 913 2875 a(implies) p Fi 1276 2843 a(e) p Fn 1256 2875 a(T) p Fo 1370 2875 a(is) p 1479 2875 a(isomorphism,) p 2091 2875 a(b) m(y) p 2237 2875 a(the) p 2416 2875 a(op) s(en) p 2663 2875 a(mapping) p 3076 2875 a(theorem) p 3466 2875 a(\(see) p 3673 2875 a([Ru2,) 0 2996 y(Theorem) p 412 2996 a(2.11]\).) p 721 2996 a(Th) m(us) p 968 2996 a(w) m(e) p 1112 2996 a(ha) m(v) m(e) 801 3228 y(dim) p Fn 980 3228 a(D) p Fo 1064 3228 a(\() p Fn(L) p Fg 1168 3187 a(\003) p Fm 1168 3253 a(N) p Fo 1235 3228 a(\)) p Fn(=D) p Fo 1406 3228 a(\() p 1444 3150 134 4 v Fn(L) p Fm 1510 3243 a(N) p Fo 1577 3228 a(\)) p 1643 3228 a(=) p Fn 1747 3228 a(N) p Fo 1852 3228 a(dim) p Fn 2031 3228 a(D) p Fo 2115 3228 a(\(\() p Fn(L) p Fm 2257 3174 a(\013) p Fe 2302 3184 a(j) p Fl 2257 3250 a(1) p Fo 2339 3228 a(\)) p Fg 2377 3187 a(\003) p Fo 2416 3228 a(\)) p Fn(=D) p Fo 2587 3228 a(\() p 2625 3135 149 4 v Fn(L) p Fm 2691 3174 a(\013) p Fe 2736 3184 a(j) p Fl 2691 3250 a(1) p Fo 2773 3228 a(\)) p 2838 3228 a(=) p 2942 3228 a(4) p Fn(N) p 3074 3228 a(;) p Fo 0 3460 a(since) p 239 3460 a(dim) p Fn 418 3460 a(D) p Fo 502 3460 a(\(\() p Fn(L) p Fm 644 3406 a(\013) p Fe 689 3416 a(j) p Fl 644 3482 a(1) p Fo 726 3460 a(\)) p Fg 764 3424 a(\003) p Fo 804 3460 a(\)) p Fn(=D) p Fo 975 3460 a(\() p 1013 3367 V Fn(L) p Fm 1079 3406 a(\013) p Fe 1124 3416 a(j) p Fl 1079 3482 a(1) p Fo 1160 3460 a(\)) p 1226 3460 a(=) p 1329 3460 a(4) p 1411 3460 a(b) m(y) p 1546 3460 a(\(ii\)) p 1709 3460 a(of) p 1820 3460 a(Prop) s(osition) p 2345 3460 a(3.2.) p 2540 3460 a(Th) m(us) p 2787 3460 a(\(4.19\)) p 3069 3460 a(holds.) p 3362 3392 64 4 v 3362 3485 4 93 v 3422 3485 V 3362 3488 64 4 v Fh 0 3793 a(5) p 242 3793 a(Characterization) p 1472 3793 a(of) p 1656 3793 a(self-adjoin) l(t) p 2486 3793 a(extensions) p Fo 0 4012 a(In) p 131 4012 a(this) p 329 4012 a(section,) p 693 4012 a(w) m(e) p 845 4012 a(shall) p 1082 4012 a(giv) m(e) p 1291 4012 a(a) p 1381 4012 a(c) m(haracterization) p 2102 4012 a(of) p 2222 4012 a(self-adjoin) m(t) p 2733 4012 a(extensions) p 3211 4012 a(of) p Fn 3331 4012 a(L) p Fm 3397 4027 a(N) p Fo 3465 4012 a(.) p 3561 4012 a(F) p 3617 4012 a(or) p 3744 4012 a(this) 0 4132 y(purp) s(ose,) p 393 4132 a(w) m(e) p 537 4132 a(shall) p 765 4132 a(in) m(tro) s(duce) p 1199 4132 a(some) p 1443 4132 a(basis) p 1683 4132 a(functions.) p Fq 146 4336 a(Lemma) p 543 4336 a(5.1) p Ff 735 4336 a(\(i\)) p 880 4336 a(L) p 936 4336 a(et) p Fn 1048 4336 a(N) p Fo 1164 4336 a(=) p 1268 4336 a(1) p Fn(;) p Fo 1361 4336 a(2) p Fn(;) p 1454 4336 a(:) p 1498 4336 a(:) p 1542 4336 a(:) p Ff 1602 4336 a(or) p Fn 1728 4336 a(N) p Fo 1844 4336 a(=) p Fk 1948 4336 a(1) p Ff(.) p 2122 4336 a(Put) p Fn 1138 4569 a(\036) p Fl 1196 4518 a(\() p Fm(j) p Fl 1256 4518 a(\)) p Fg 1196 4591 a(\000) p Fl(1) p Fo 1373 4569 a(=) p Fn 1532 4569 a(t) p Fm 1567 4584 a(z) p Fe 1600 4594 a(j) p Fo 1637 4569 a(\() p Fn(\037\036) p Fm 1794 4515 a(\013) p Fe 1839 4525 a(j) p Fg 1794 4591 a(\000) p Fl(1) p Fo 1888 4569 a(\)) p Fn(;) p 2005 4569 a( ) p Fl 2072 4518 a(\() p Fm(j) p Fl 2132 4518 a(\)) 2068 4591 y(1) p Fo 2191 4569 a(=) p Fn 2294 4569 a(t) p Fm 2329 4584 a(z) p Fe 2362 4594 a(j) p Fo 2399 4569 a(\() p Fn(\037 ) p Fm 2565 4515 a(\013) p Fe 2610 4525 a(j) p Fl 2561 4591 a(1) p Fo 2647 4569 a(\)) p Fn(;) 1141 4738 y(\036) p Fl 1199 4687 a(\() p Fm(j) p Fl 1259 4687 a(\)) 1199 4760 y(0) p Fo 1373 4738 a(=) p Fn 1532 4738 a(t) p Fm 1567 4753 a(z) p Fe 1600 4763 a(j) p Fo 1637 4738 a(\() p Fn(\037\036) p Fm 1794 4684 a(\013) p Fe 1839 4694 a(j) p Fl 1794 4760 a(0) p Fo 1876 4738 a(\)) p Fn(;) p 1992 4738 a( ) p Fl 2059 4687 a(\() p Fm(j) p Fl 2119 4687 a(\)) 2055 4760 y(0) p Fo 2178 4738 a(=) p Fn 2282 4738 a(t) p Fm 2317 4753 a(z) p Fe 2350 4763 a(j) p Fo 2387 4738 a(\() p Fn(\037 ) p Fm 2553 4684 a(\013) p Fe 2598 4694 a(j) p Fl 2549 4760 a(0) p Fo 2635 4738 a(\)) 1902 5214 y(26) p eop %%Page: 27 27 27 26 bop Ff 0 631 a(for) p Fn 147 631 a(j) p Fo 220 631 a(=) p 324 631 a(1) p Fn(;) p 417 631 a(:) p 461 631 a(:) p 505 631 a(:) p 547 631 a(;) p 591 631 a(N) p Ff 679 631 a(,) p 737 631 a(wher) p 933 631 a(e) p Fn 1003 631 a(\037) p Ff 1090 631 a(is) p 1185 631 a(a) p 1261 631 a(function) p 1635 631 a(given) p 1880 631 a(in) p 1991 631 a(L) p 2047 631 a(emma) p 2327 631 a(4.4) p 2482 631 a(and) p Fn 2662 631 a(t) p Fm 2697 646 a(z) p Fe 2730 656 a(j) p Ff 2792 631 a(is) p 2888 631 a(the) p 3041 631 a(magnetic) p 3447 631 a(tr) p 3515 631 a(anslation) 0 751 y(op) p 95 751 a(er) p 176 751 a(ator) p 377 751 a(fr) p 443 751 a(om) p Fk 601 751 a(fj) p Fn(z) p Fk 728 751 a(j) p Fn 783 751 a(<) p 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1191 a(Then,) p 284 1191 a(the) p 446 1191 a(fol) p 556 1191 a(lowing) p 859 1191 a(e) p 899 1191 a(qualities) p 1283 1191 a(hold:) p Fo 987 1421 a([) p Fn(\036) p Fl 1072 1370 a(\() p Fm(j) p Fl 1132 1370 a(\)) p Fm 1072 1446 a(l) p Fn 1163 1421 a(;) p 1207 1421 a(\036) p Fl 1265 1370 a(\() p Fm(j) p Fb 1325 1346 a(0) p Fl 1347 1370 a(\)) p Fm 1265 1446 a(l) p Fb 1287 1427 a(0) p Fo 1379 1421 a(]) p Fm 1406 1436 a(N) p Fo 1501 1421 a(=) p 1604 1421 a([) p Fn(\036) p Fl 1689 1370 a(\() p Fm(j) p Fl 1749 1370 a(\)) p Fm 1689 1446 a(l) p Fn 1780 1421 a(;) p 1824 1421 a( ) p Fl 1891 1380 a(\() p Fm(j) p Fb 1951 1356 a(0) p Fl 1973 1380 a(\)) p Fm 1887 1445 a(n) p Fo 2005 1421 a(]) p Fm 2032 1436 a(N) p Fo 2127 1421 a(=) p 2230 1421 a([) p Fn( ) p Fl 2324 1380 a(\() p Fm(j) p Fl 2384 1380 a(\)) p Fm 2320 1445 a(n) p Fn 2416 1421 a(;) p 2460 1421 a( ) p Fl 2527 1370 a(\() p Fm(j) p Fb 2587 1346 a(0) p Fl 2609 1370 a(\)) p Fm 2523 1446 a(n) p Fb 2566 1427 a(0) p Fo 2640 1421 a(]) p Fm 2667 1436 a(N) p Fo 2762 1421 a(=) p 2866 1421 a(0) p 3701 1421 a(\(5.1\)) p Ff 0 1641 a(for) p Fn 156 1641 a(j) p Fk 229 1641 a(6) p Fo(=) p Fn 333 1641 a(j) p Fg 379 1605 a(0) p Ff 402 1641 a(,) p Fn 467 1641 a(l) r(;) p 542 1641 a(l) p Fg 573 1605 a(0) p Fo 624 1641 a(=) p Fk 728 1641 a(\000) p Fo(1) p Fn(;) p Fo 898 1641 a(0) p Ff(,) p Fn 1011 1641 a(n;) p 1113 1641 a(n) p Fg 1171 1605 a(0) p Fo 1222 1641 a(=) p 1326 1641 a(0) p Fn(;) p Fo 1419 1641 a(1) p Ff(.) p Fo 777 1861 a([) p Fn(\036) p Fl 862 1810 a(\() p Fm(j) p Fl 922 1810 a(\)) p Fg 862 1882 a(\000) p Fl(1) p Fn 956 1861 a(;) p 1000 1861 a( ) p Fl 1067 1810 a(\() p Fm(j) p Fl 1127 1810 a(\)) 1063 1882 y(1) p Fo 1158 1861 a(]) p Fm 1185 1876 a(N) p Fo 1280 1861 a(=) p 1384 1861 a(4) p Fn(\031) p Fo 1492 1861 a(\() p Fn(\013) p Fm 1592 1876 a(j) p Fk 1650 1861 a(\000) p Fo 1750 1861 a(1\)) p Fn(;) p Fo 1881 1861 a([) p Fn(\036) p Fl 1966 1810 a(\() p Fm(j) p Fl 2026 1810 a(\)) 1966 1882 y(0) p Fn 2056 1861 a(;) p 2100 1861 a( ) p Fl 2167 1810 a(\() p Fm(j) p Fl 2227 1810 a(\)) 2163 1882 y(0) p Fo 2259 1861 a(]) p Fm 2286 1876 a(N) p Fo 2381 1861 a(=) p 2484 1861 a(4) p Fn(\031) t(\013) p Fm 2654 1876 a(j) p Fn 2690 1861 a(;) p Fo 3701 1861 a(\(5.2\)) 777 2023 y([) p Fn(\036) p Fl 862 1972 a(\() p Fm(j) p Fl 922 1972 a(\)) p Fg 862 2045 a(\000) p Fl(1) p Fn 956 2023 a(;) p 1000 2023 a(\036) p Fl 1058 1972 a(\() p Fm(j) p Fl 1118 1972 a(\)) 1058 2045 y(0) p Fo 1149 2023 a(]) p Fm 1176 2038 a(N) p Fo 1271 2023 a(=) p 1375 2023 a([) p Fn(\036) p Fl 1460 1972 a(\() p Fm(j) p Fl 1520 1972 a(\)) p Fg 1460 2045 a(\000) p Fl(1) p Fn 1554 2023 a(;) p 1598 2023 a( ) p Fl 1665 1972 a(\() p Fm(j) p Fl 1725 1972 a(\)) 1661 2045 y(0) p Fo 1756 2023 a(]) p Fm 1783 2038 a(N) p Fo 1878 2023 a(=) p 1981 2023 a([) p Fn( ) p Fl 2075 1972 a(\() p Fm(j) p Fl 2135 1972 a(\)) 2071 2045 y(1) p Fn 2167 2023 a(;) p 2211 2023 a(\036) p Fl 2269 1972 a(\() p Fm(j) p Fl 2329 1972 a(\)) 2269 2045 y(0) p Fo 2360 2023 a(]) p Fm 2387 2038 a(N) p Fo 2482 2023 a(=) p 2585 2023 a([) p Fn( ) p Fl 2679 1972 a(\() p Fm(j) p Fl 2739 1972 a(\)) 2675 2045 y(1) p Fn 2771 2023 a(;) p 2815 2023 a( ) p Fl 2882 1972 a(\() p Fm(j) p Fl 2942 1972 a(\)) 2878 2045 y(0) p Fo 2973 2023 a(]) p Fm 3000 2038 a(N) p Fo 3095 2023 a(=) p 3198 2023 a(0) p Fn(;) p Fo 777 2186 a([) p Fn(\036) p Fl 862 2135 a(\() p Fm(j) p Fl 922 2135 a(\)) p Fm 862 2211 a(l) p Fn 953 2186 a(;) p 997 2186 a(\036) p Fl 1055 2135 a(\() p Fm(j) p Fl 1115 2135 a(\)) p Fm 1055 2211 a(l) p Fo 1146 2186 a(]) p Fm 1173 2201 a(N) p Fo 1268 2186 a(=) p 1372 2186 a([) p Fn( ) p Fl 1466 2145 a(\() p Fm(j) p Fl 1526 2145 a(\)) p Fm 1462 2211 a(n) p Fn 1557 2186 a(;) p 1601 2186 a( ) p Fl 1668 2145 a(\() p Fm(j) p Fl 1728 2145 a(\)) p Fm 1664 2211 a(n) p Fo 1759 2186 a(]) p Fm 1786 2201 a(N) p Fo 1881 2186 a(=) p 1985 2186 a(0) p 2138 2186 a(for) p Fn 2290 2186 a(l) p Fo 2349 2186 a(=) p Fk 2452 2186 a(\000) p Fo(1) p Fn(;) p Fo 2622 2186 a(0) p Fn(;) p 2749 2186 a(n) p Fo 2835 2186 a(=) p 2939 2186 a(0) p Fn(;) p Fo 3032 2186 a(1) p Ff 0 2406 a(for) p Fn 156 2406 a(j) p Fo 229 2406 a(=) p 333 2406 a(1) p Fn(;) p 426 2406 a(:) p 470 2406 a(:) p 514 2406 a(:) p 556 2406 a(;) p 600 2406 a(N) p Ff 688 2406 a(.) 146 2526 y(\(ii\)) p 321 2526 a(De\014ne) p 628 2526 a(line) p 778 2526 a(ar) p 904 2526 a(functionals) p 1402 2526 a(on) p Fn 1541 2526 a(D) p Fo 1625 2526 a(\() p Fn(L) p Fg 1729 2490 a(\003) p Fm 1729 2551 a(N) p Fo 1796 2526 a(\)) p Ff 1869 2526 a(by) p Fo 1067 2760 a(\010) p Fl 1137 2709 a(\() p Fm(j) p Fl 1197 2709 a(\)) p Fg 1137 2782 a(\000) p Fl(1) p Fo 1232 2760 a(\() p Fn(u) p Fo(\)) p 1391 2760 a(=) p 1495 2760 a(\010) p Fm 1565 2706 a(\013) p Fe 1610 2716 a(j) p Fg 1565 2782 a(\000) p Fl(1) p Fo 1660 2760 a(\() p Fn(T) p Fm 1755 2775 a(j) p Fn 1791 2760 a(u) p Fo(\)) p Fn(;) p Fo 1995 2760 a(\011) p Fl 2071 2709 a(\() p Fm(j) p Fl 2131 2709 a(\)) 2071 2782 y(1) p Fo 2162 2760 a(\() p Fn(u) p Fo(\)) p 2321 2760 a(=) p 2425 2760 a(\011) p Fm 2501 2706 a(\013) p Fe 2546 2716 a(j) p Fl 2501 2782 a(1) p Fo 2582 2760 a(\() p Fn(T) p Fm 2677 2775 a(j) p Fn 2714 2760 a(u) p Fo(\)) p Fn(;) p Fo 1075 2897 a(\010) p Fl 1145 2846 a(\() p Fm(j) p Fl 1205 2846 a(\)) 1145 2919 y(0) p Fo 1237 2897 a(\() p Fn(u) p Fo(\)) p 1396 2897 a(=) p 1499 2897 a(\010) p Fm 1569 2843 a(\013) p Fe 1614 2853 a(j) p Fl 1569 2919 a(0) p Fo 1652 2897 a(\() p Fn(T) p Fm 1747 2912 a(j) p Fn 1784 2897 a(u) p Fo(\)) p Fn(;) p Fo 1995 2897 a(\011) p Fl 2071 2846 a(\() p Fm(j) p Fl 2131 2846 a(\)) 2071 2919 y(0) p Fo 2162 2897 a(\() p Fn(u) p Fo(\)) p 2321 2897 a(=) p 2425 2897 a(\011) p Fm 2501 2843 a(\013) p Fe 2546 2853 a(j) p Fl 2501 2919 a(0) p Fo 2582 2897 a(\() p Fn(T) p Fm 2677 2912 a(j) p Fn 2714 2897 a(u) p Fo(\)) p Fn(;) p Fo 3701 2821 a(\(5.3\)) p Ff 0 3116 a(for) p Fn 166 3116 a(j) p Fo 259 3116 a(=) p 382 3116 a(1) p Fn(;) p 475 3116 a(:) p 519 3116 a(:) p 563 3116 a(:) p 605 3116 a(;) p 649 3116 a(N) p Ff 783 3116 a(and) p Fn 982 3116 a(u) p Fk 1085 3116 a(2) p Fn 1198 3116 a(D) p Fo 1282 3116 a(\() p Fn(L) p Fg 1386 3079 a(\003) p Fm 1386 3140 a(N) p Fo 1453 3116 a(\)) p Ff(,) p 1569 3116 a(wher) p 1765 3116 a(e) p Fn 1855 3116 a(T) p Fm 1912 3131 a(j) p Ff 1994 3116 a(is) p 2109 3116 a(the) p 2281 3116 a(op) p 2376 3116 a(er) p 2457 3116 a(ator) p 2675 3116 a(de\014ne) p 2920 3116 a(d) p 3014 3116 a(by) p 3152 3116 a(\() p Fo(4) p Fn(:) p Fo(10) p Ff(\).) p 3511 3116 a(Then) p 3775 3116 a(the) 0 3236 y(fol) p 110 3236 a(lowing) p 413 3236 a(e) p 453 3236 a(qualities) p 837 3236 a(hold:) p Fo 1460 3356 a(\010) p Fl 1530 3305 a(\() p Fm(j) p Fl 1590 3305 a(\)) p Fm 1530 3381 a(l) p Fo 1622 3356 a(\() p Fn(u) p Fl 1716 3371 a(0) p Fo 1755 3356 a(\)) p 1820 3356 a(=) p 1924 3356 a(\011) p Fl 2000 3315 a(\() p Fm(j) p Fl 2060 3315 a(\)) p Fm 2000 3381 a(n) p Fo 2091 3356 a(\() p Fn(u) p Fl 2185 3371 a(0) p Fo 2224 3356 a(\)) p 2290 3356 a(=) p 2393 3356 a(0) p 3701 3356 a(\(5.4\)) p Ff 0 3531 a(for) p Fn 156 3531 a(j) p Fo 229 3531 a(=) p 333 3531 a(1) p Fn(;) p 426 3531 a(:) p 470 3531 a(:) p 514 3531 a(:) p 556 3531 a(;) p 600 3531 a(N) p Ff 688 3531 a(,) p Fn 753 3531 a(l) p Fo 812 3531 a(=) p Fk 916 3531 a(\000) p Fo(1) p Fn(;) p Fo 1086 3531 a(0) p Ff(,) p Fn 1199 3531 a(n) p Fo 1285 3531 a(=) p 1389 3531 a(0) p Fn(;) p Fo 1482 3531 a(1) p Ff 1565 3531 a(and) p Fn 1754 3531 a(u) p Fl 1810 3546 a(0) p Fk 1877 3531 a(2) p Fn 1971 3531 a(D) p Fo 2055 3531 a(\() p 2093 3453 134 4 v Fn(L) p Fm 2159 3546 a(N) p Fo 2226 3531 a(\)) p Ff(.) p Fo 817 3766 a(\010) p Fl 887 3715 a(\() p Fm(j) p Fl 947 3715 a(\)) p Fm 887 3791 a(l) p Fo 979 3766 a(\() p Fn(\036) p Fl 1075 3715 a(\() p Fm(j) p Fb 1135 3691 a(0) p Fl 1157 3715 a(\)) p Fm 1075 3791 a(l) p Fb 1097 3772 a(0) p Fo 1189 3766 a(\)) p 1254 3766 a(=) p 1358 3766 a(\011) p Fl 1434 3724 a(\() p Fm(j) p Fl 1494 3724 a(\)) p Fm 1434 3790 a(n) p Fo 1525 3766 a(\() p Fn( ) p Fl 1630 3715 a(\() p Fm(j) p Fb 1690 3691 a(0) p Fl 1712 3715 a(\)) p Fm 1626 3791 a(n) p Fb 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1485 3986 a(0) p Fn(;) p Fo 1578 3986 a(1) p Ff(.) p Fo 583 4206 a(\010) p Fl 653 4155 a(\() p Fm(j) p Fl 713 4155 a(\)) p Fm 653 4231 a(l) p Fo 745 4206 a(\() p Fn(\036) p Fl 841 4155 a(\() p Fm(j) p Fl 901 4155 a(\)) p Fm 841 4231 a(l) p Fb 863 4212 a(0) p Fo 932 4206 a(\)) p 998 4206 a(=) p Fn 1101 4206 a(\016) p Fm 1144 4221 a(l) q(;l) p Fb 1208 4202 a(0) p Fn 1235 4206 a(;) p Fo 1313 4206 a(\011) p Fl 1389 4164 a(\() p Fm(j) p Fl 1449 4164 a(\)) p Fm 1389 4230 a(n) p Fo 1480 4206 a(\() p Fn( ) p Fl 1585 4155 a(\() p Fm(j) p Fl 1645 4155 a(\)) p Fm 1581 4231 a(n) p Fb 1624 4212 a(0) p Fo 1677 4206 a(\)) p 1742 4206 a(=) p Fn 1846 4206 a(\016) p Fm 1889 4221 a(n;n) p Fb 1995 4202 a(0) p Fn 2021 4206 a(;) p Fo 2065 4206 a(\010) p Fl 2135 4155 a(\() p Fm(j) p Fl 2195 4155 a(\)) p Fm 2135 4231 a(l) p Fo 2226 4206 a(\() p Fn( ) p Fl 2331 4155 a(\() p Fm(j) p Fl 2391 4155 a(\)) p Fm 2327 4231 a(l) p Fb 2349 4212 a(0) p Fo 2423 4206 a(\)) p 2488 4206 a(=) p 2592 4206 a(0) p Fn(;) p Fo 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4688 a(\016) p Fm 1507 4703 a(m;n) p Fo 1664 4688 a(=) p Fi 1768 4567 a(\032) p Fo 1846 4628 a(1) p Ff 1995 4628 a(\() p Fn(m) p Fo 2148 4628 a(=) p Fn 2251 4628 a(n) p Ff(\)) p Fo 1846 4748 a(0) p Ff 1995 4748 a(\() p Fn(m) p Fk 2148 4748 a(6) p Fo(=) p Fn 2251 4748 a(n) p Ff 2344 4748 a(\)) p Fn 2411 4688 a(:) p Fo 1902 5214 a(27) p eop %%Page: 28 28 28 27 bop Ff 146 631 a(Pr) p 248 631 a(o) p 293 631 a(of.) p Fo 443 631 a(\(i\)) p 586 631 a(The) p 794 631 a(equalities) p 1233 631 a(\(5.1\)) p 1473 631 a(clearly) p 1792 631 a(hold) p 2016 631 a(since) p 2263 631 a(the) p 2438 631 a(in) m(tersection) p 2969 631 a(of) p 3088 631 a(the) p 3263 631 a(supp) s(ort) p 3632 631 a(of) p 3750 631 a(t) m(w) m(o) 0 751 y(functions) p 421 751 a(in) p 534 751 a(eac) m(h) p 754 751 a(brac) m(k) m(et) p 1098 751 a(is) p 1196 751 a(empt) m(y) p 1453 751 a(.) p 1524 751 a(The) p 1724 751 a(\014rst) p 1926 751 a(equalit) m(y) p 2297 751 a(of) p 2408 751 a(\(5.2\)) p 2641 751 a(is) p 2739 751 a(pro) 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2882 a(can) p 2891 2882 a(b) s(e) p 3024 2882 a(pro) m(v) m(ed) p 3341 2882 a(similarly) p 3703 2882 a(.) p 3769 2815 64 4 v 3769 2907 4 93 v 3829 2907 V 3769 2910 64 4 v 146 3100 a(W) p 238 3100 a(e) p 314 3100 a(shall) p 542 3100 a(determine) p 998 3100 a(the) p 1166 3100 a(structure) p 1584 3100 a(of) p Fn 1695 3100 a(D) p Fo 1779 3100 a(\() p Fn(L) p Fg 1883 3064 a(\003) p Fm 1883 3125 a(N) p Fo 1950 3100 a(\)) p 2020 3100 a(under) p 2297 3100 a(the) p 2465 3100 a(follo) m(wing) p 2877 3100 a(assumption.) p Fq 146 3295 a(Assumption) p 768 3295 a(1) p Ff 873 3295 a(Ther) p 1074 3295 a(e) p 1153 3295 a(exist) p 1379 3295 a(c) p 1419 3295 a(onstants) p Fn 1808 3295 a(\013) p Fg 1870 3310 a(\000) p Ff 1929 3295 a(,) p Fn 1994 3295 a(\013) p Fl 2056 3310 a(+) p Ff 2150 3295 a(such) p 2371 3295 a(that) p Fo 969 3506 a(0) p Fn 1046 3506 a(<) p 1149 3506 a(\013) p Fg 1211 3521 a(\000) p Fk 1298 3506 a(\024) p Fn 1403 3506 a(\013) p Fm 1465 3521 a(j) p Fk 1529 3506 a(\024) p Fn 1634 3506 a(\013) p Fl 1696 3521 a(+) p Fn 1783 3506 a(<) p Fo 1887 3506 a(1) p 1970 3506 a(for) p 2119 3506 a(ev) m(ery) p Fn 2378 3506 a(j) p Fo 2452 3506 a(=) p 2555 3506 a(1) p Fn(;) p 2648 3506 a(:) p 2692 3506 a(:) p 2736 3506 a(:) p 2779 3506 a(;) p 2823 3506 a(N) p 2906 3506 a(:) p Fo 0 3717 a(Remark) p 367 3717 a(1.) p 486 3717 a(This) p 709 3717 a(assumption) p 1225 3717 a(is) p 1323 3717 a(satis\014ed) p 1698 3717 a(when) p Fn 1953 3717 a(N) p Fo 2074 3717 a(is) p 2172 3717 a(\014nite.) 0 3837 y(Remark) p 378 3837 a(2.) p 529 3837 a(It) p 646 3837 a(is) p 755 3837 a(kno) m(wn) p 1074 3837 a(that) p 1297 3837 a(the) p 1475 3837 a(de\014ciency) p 1933 3837 a(indices) p 2264 3837 a(of) p Fn 2386 3837 a(L) p Fl 2452 3801 a(0) 2452 3862 y(1) p Fo 2538 3837 a(=) p Fk 2660 3837 a(\000) p Fo(\001) p Fk(j) p Fm 2846 3854 a(C) p Fb 2901 3831 a(1) p Fd 2896 3876 a(0) p Fl 2966 3854 a(\() p Fa(R) p Fd 3054 3835 a(2) p Fg 3089 3854 a(nf) p Fl(0) p Fg(g) p Fl(\)) p Fo 3305 3837 a(are) p 3478 3837 a(\(1) p Fn(;) p Fo 3609 3837 a(1\)) p 3739 3837 a(\(see) 0 3957 y([Al-Ge-Ho-Ho]\).) p 740 3957 a(Since) p 1000 3957 a(the) p 1174 3957 a(op) s(erator) p Fn 1572 3957 a(L) p Fl 1638 3921 a(1) 1638 3982 y(1) p Fo 1716 3957 a(is) p 1819 3957 a(unitarily) p 2223 3957 a(equiv) p 2445 3957 a(alen) m(t) p 2692 3957 a(to) p Fn 2816 3957 a(L) p Fl 2882 3921 a(0) 2882 3982 y(1) p Fo 2922 3957 a(,) p 2988 3957 a(the) p 3162 3957 a(de\014ciency) p 3614 3957 a(indices) 0 4078 y(of) p Fn 115 4078 a(L) p Fl 181 4041 a(1) 181 4102 y(1) p Fo 257 4078 a(are) p 423 4078 a(also) p 623 4078 a(\(1) p Fn(;) p Fo 754 4078 a(1\).) p 922 4078 a(Th) m(us) p 1173 4078 a(the) p 1345 4078 a(de\014ciency) p 1796 4078 a(indices) p 2120 4078 a(of) p Fn 2235 4078 a(L) p Fm 2301 4041 a(\013) p Fl 2301 4102 a(1) p Fo 2387 4078 a(c) m(hange) p 2713 4078 a(discon) m(tin) m(uously) p 3404 4078 a(at) p Fn 3527 4078 a(\013) p Fo 3624 4078 a(=) p 3734 4078 a(0) p Fn(;) p Fo 3827 4078 a(1.) 0 4198 y(Hence) p 296 4198 a(some) p 547 4198 a(of) p 665 4198 a(our) p 845 4198 a(estimates) p 1280 4198 a(ha) m(v) m(e) p 1512 4198 a(singularities) p 2061 4198 a(at) p Fn 2187 4198 a(\013) p Fo 2288 4198 a(=) p 2403 4198 a(0) p Fn(;) p Fo 2496 4198 a(1) p 2583 4198 a(\(for) p 2776 4198 a(example,) p 3194 4198 a(see) p 3358 4198 a(Lemma) p 3712 4198 a(3.3\).) 0 4318 y(W) p 92 4318 a(e) p 168 4318 a(imp) s(ose) p 496 4318 a(Assumption) p 1037 4318 a(1) p 1118 4318 a(to) p 1237 4318 a(a) m(v) m(oid) p 1495 4318 a(these) p 1744 4318 a(singularities.) 146 4439 y(In) p 275 4439 a(the) p 450 4439 a(sequel,) p Fj 777 4439 a(C) p Fl 865 4397 a(4) p Fm(N) p Fo 1007 4439 a(denotes) p 1367 4439 a(the) p 1542 4439 a(Hilb) s(ert) p 1885 4439 a(space) p 2153 4439 a(of) p 2271 4439 a(the) p 2446 4439 a(sequence) p 2857 4439 a(of) p 2975 4439 a(4) p Fn(N) p Fo 3152 4439 a(complex) p 3541 4439 a(n) m(um) m(b) s(ers) 0 4559 y(\(in\014nite) p 369 4559 a(complex) p 751 4559 a(n) m(um) m(b) s(ers,) p 1171 4559 a(when) p Fn 1426 4559 a(N) p Fo 1542 4559 a(=) p Fk 1646 4559 a(1) p Fo(\)) p 1816 4559 a(equipp) s(ed) p 2233 4559 a(with) p 2455 4559 a(the) p 2623 4559 a(norm) p Fk 1171 4844 a(jj) p Fq(c) p Fk(jj) p Fl 1333 4803 a(2) p Fa 1333 4869 a(C) p Fd 1392 4850 a(4) p Fe(N) p Fo 1510 4844 a(=) p Fl 1626 4736 a(4) p Fm(N) p Fi 1615 4761 a(X) p Fm 1614 4943 a(j) p Fl 1647 4943 a(=1) p Fk 1753 4844 a(j) p Fn(c) p Fm 1823 4859 a(j) p Fk 1859 4844 a(j) p Fl 1887 4803 a(2) p Fn 1926 4844 a(;) p Fq 2002 4844 a(c) p Fo 2080 4844 a(=) p 2184 4844 a(\() p Fn(c) p Fl 2264 4859 a(1) p Fn 2303 4844 a(;) p 2347 4844 a(:) p 2391 4844 a(:) p 2435 4844 a(:) p 2478 4844 a(;) p 2522 4844 a(c) p Fl 2564 4859 a(4) p Fm(N) p Fo 2666 4844 a(\)) p Fn(:) p Fo 1902 5214 a(28) p eop %%Page: 29 29 29 28 bop Fq 146 631 a(Lemma) p 543 631 a(5.2) p Ff 735 631 a(L) p 791 631 a(et) p Fn 915 631 a(N) p Fo 1052 631 a(=) p 1176 631 a(1) p Fn(;) p Fo 1269 631 a(2) p Fn(;) p 1362 631 a(:) p 1406 631 a(:) p 1450 631 a(:) p Ff 1522 631 a(or) p Fn 1659 631 a(N) p Fo 1796 631 a(=) p Fk 1921 631 a(1) p Ff(.) p 2129 631 a(Supp) p 2331 631 a(ose) p 2511 631 a(Assumption) p 3058 631 a(1) p 3154 631 a(holds.) p 3477 631 a(Then,) p 3775 631 a(the) 0 751 y(fol) p 110 751 a(lowing) p 413 751 a(holds.) 146 872 y(\(i\)) p 291 872 a(F) p 348 872 a(or) p Fn 473 872 a(u) p Fk 556 872 a(2) p Fn 650 872 a(D) p Fo 734 872 a(\() p Fn(L) p Fg 838 836 a(\003) p Fm 838 896 a(N) p Fo 905 872 a(\)) p Ff(,) p 1008 872 a(the) p 1170 872 a(se) p 1250 872 a(quenc) p 1487 872 a(e) p Fo 944 1076 a(\004) p Fn(u) p Fo 1148 1076 a(=) p 1307 1076 a(\(\010) p Fl 1415 1025 a(\(1\)) p Fg 1415 1098 a(\000) p Fl(1) p Fo 1510 1076 a(\() p Fn(u) p Fo(\)) p Fn(;) p Fo 1686 1076 a(\011) p Fl 1762 1025 a(\(1\)) 1762 1098 y(1) p Fo 1855 1076 a(\() p Fn(u) p Fo(\)) p Fn(;) p Fo 2031 1076 a(\010) p Fl 2101 1025 a(\(1\)) 2101 1098 y(0) p Fo 2195 1076 a(\() p Fn(u) p Fo(\)) p Fn(;) p Fo 2371 1076 a(\011) p Fl 2447 1025 a(\(1\)) 2447 1098 y(0) p Fo 2540 1076 a(\() p Fn(u) p Fo(\)) p Fn(;) p 2716 1076 a(:) p 2760 1076 a(:) p 2804 1076 a(:) 1307 1238 y(:) p 1351 1238 a(:) p 1395 1238 a(:) p 1438 1238 a(;) p Fo 1482 1238 a(\010) p Fl 1552 1188 a(\() p Fm(N) p Fl 1642 1188 a(\)) p Fg 1552 1260 a(\000) p Fl(1) p Fo 1674 1238 a(\() p Fn(u) p Fo(\)) p Fn(;) p Fo 1850 1238 a(\011) p Fl 1926 1188 a(\() p Fm(N) p Fl 2016 1188 a(\)) 1926 1260 y(1) p Fo 2047 1238 a(\() p Fn(u) p Fo(\)) p Fn(;) p Fo 2223 1238 a(\010) p Fl 2293 1188 a(\() p Fm(N) p Fl 2383 1188 a(\)) 2293 1260 y(0) p Fo 2415 1238 a(\() p Fn(u) p Fo(\)) p Fn(;) p Fo 2591 1238 a(\011) p Fl 2667 1188 a(\() p Fm(N) p Fl 2757 1188 a(\)) 2667 1260 y(0) p Fo 2788 1238 a(\() p Fn(u) p Fo(\)\)) p 3701 1238 a(\(5.7\)) p Ff 0 1429 a(b) p 40 1429 a(elongs) p 334 1429 a(to) p Fj 451 1429 a(C) p Fl 539 1387 a(4) p Fm(N) p Ff 677 1429 a(for) p 832 1429 a(any) p Fn 1019 1429 a(u) p Fk 1102 1429 a(2) p Fn 1196 1429 a(D) p Fo 1280 1429 a(\() p Fn(L) p Fg 1384 1393 a(\003) p Fm 1384 1454 a(N) p Fo 1452 1429 a(\)) p Ff 1524 1429 a(\(when) p Fn 1813 1429 a(N) p Fo 1929 1429 a(=) p Fk 2033 1429 a(1) p Ff(,) p Fo 2197 1429 a(\004) p Fn(u) p Ff 2353 1429 a(is) p 2457 1429 a(an) p 2597 1429 a(in\014nite) p 2933 1429 a(se) p 3013 1429 a(quenc) p 3250 1429 a(e\).) p 3409 1429 a(The) p 3608 1429 a(map) p Fn 1422 1620 a(D) p Fo 1506 1620 a(\() p Fn(L) p Fg 1610 1578 a(\003) p Fm 1610 1644 a(N) p Fo 1677 1620 a(\)) p Fk 1743 1620 a(3) p Fn 1837 1620 a(u) p Fk 1920 1620 a(7!) p Fo 2047 1620 a(\004) p Fn(u) p Fk 2195 1620 a(2) p Fj 2289 1620 a(C) p Fl 2378 1578 a(4) p Fm(N) p Ff 0 1810 a(is) p 101 1810 a(c) p 141 1810 a(ontinuous.) p 630 1810 a(Mor) p 803 1810 a(e) p 843 1810 a(over,) p 1086 1810 a(ther) p 1249 1810 a(e) p 1325 1810 a(exist) p 1548 1810 a(a) p 1629 1810 a(c) p 1669 1810 a(onstant) p Fn 2014 1810 a(C) p Fl 2084 1825 a(2) p Ff 2155 1810 a(dep) p 2295 1810 a(endent) p 2607 1810 a(on) p Fn 2743 1810 a(B) p Ff 2822 1810 a(,) p Fn 2884 1810 a(R) p Ff 2959 1810 a(,) p Fn 3021 1810 a(\013) p Fg 3083 1825 a(\000) p Ff 3142 1810 a(,) p Fn 3204 1810 a(\013) p Fl 3266 1825 a(+) p Ff 3357 1810 a(\(indep) p 3622 1810 a(endent) 0 1931 y(of) p Fn 115 1931 a(N) p Ff 203 1931 a(\)) p 278 1931 a(such) p 499 1931 a(that) p Fk 1522 2051 a(jj) p Fo(\004) p Fn(u) p Fk(jj) p Fl 1755 2010 a(2) p Fa 1755 2076 a(C) p Fd 1814 2057 a(4) p Fe(N) p Fk 1932 2051 a(\024) p Fn 2037 2051 a(C) p Fl 2107 2066 a(2) p Fk 2146 2051 a(jj) p Fn(u) p Fk(jj) p Fl 2314 2010 a(2) p Fm 2314 2076 a(N) p Fo 3701 2051 a(\(5.8\)) p Ff 0 2213 a(for) p 156 2213 a(any) p Fn 342 2213 a(u) p Fk 426 2213 a(2) p Fn 520 2213 a(D) p Fo 604 2213 a(\() p Fn(L) p Fg 708 2177 a(\003) p Fm 708 2237 a(N) p Fo 775 2213 a(\)) p Ff(.) 146 2333 y(\(ii\)) p 321 2333 a(F) p 378 2333 a(or) p Fq 503 2333 a(c) p Fo 580 2333 a(=) p 684 2333 a(\() p Fn(c) p Fl 764 2348 a(1) p Fn 803 2333 a(;) p 847 2333 a(:) p 891 2333 a(:) p 935 2333 a(:) p 978 2333 a(;) p 1022 2333 a(c) p Fl 1064 2348 a(4) p Fm(N) p Fo 1166 2333 a(\)) p Fk 1232 2333 a(2) p Fj 1326 2333 a(C) p 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a(\000) p Fl(2) p Fn 1970 2700 a( ) p Fl 2037 2649 a(\() p Fm(N) p Fl 2127 2649 a(\)) 2033 2722 y(1) p Fo 2181 2700 a(+) p Fn 2279 2700 a(c) p Fl 2321 2715 a(4) p Fm(N) p Fg 2419 2715 a(\000) p Fl(1) p Fn 2514 2700 a(\036) p Fl 2572 2649 a(\() p Fm(N) p Fl 2662 2649 a(\)) 2572 2722 y(0) p Fo 2716 2700 a(+) p Fn 2814 2700 a(c) p Fl 2856 2715 a(4) p Fm(N) p Fn 2958 2700 a( ) p Fl 3025 2649 a(\() p Fm(N) p Fl 3115 2649 a(\)) 3021 2722 y(0) p Fo 3701 2700 a(\(5.9\)) p Ff 0 2890 a(b) p 40 2890 a(elongs) p 339 2890 a(to) p Fn 462 2890 a(D) p Fo 546 2890 a(\() p Fn(L) p Fg 650 2854 a(\003) p Fm 650 2915 a(N) p Fo 717 2890 a(\)) p Ff 796 2890 a(\(when) p Fn 1090 2890 a(N) p Fo 1217 2890 a(=) p Fk 1331 2890 a(1) p Ff(,) p 1502 2890 a(the) p 1670 2890 a(right) p 1908 2890 a(hand) p 2153 2890 a(side) p 2358 2890 a(of) p 2478 2890 a(\(5.9\)) p 2728 2890 a(is) p 2838 2890 a(a) p 2928 2890 a(c) p 2968 2890 a(onver) p 3199 2890 a(gent) p 3416 2890 a(se) p 3496 2890 a(quenc) p 3733 2890 a(e) p 3817 2890 a(in) p Fn 0 3011 a(D) p Fo 84 3011 a(\() p Fn(L) p Fg 188 2975 a(\003) 188 3035 y(1) p Fo 263 3011 a(\)) p Ff(\).) p 414 3011 a(Mor) p 587 3011 a(e) p 627 3011 a(over,) p 870 3011 a(ther) p 1033 3011 a(e) p 1109 3011 a(exist) p 1332 3011 a(a) p 1413 3011 a(c) p 1453 3011 a(onstant) p Fn 1798 3011 a(C) p Fl 1868 3026 a(3) p Ff 1939 3011 a(dep) p 2079 3011 a(endent) p 2391 3011 a(on) p Fn 2527 3011 a(B) p 2606 3011 a(;) p 2650 3011 a(R) q(;) p 2769 3011 a(\013) p Fg 2831 3026 a(\000) p Fn 2890 3011 a(;) p 2934 3011 a(\013) p Fl 2996 3026 a(+) p Ff 3086 3011 a(\(indep) p 3351 3011 a(endent) p 3663 3011 a(of) p Fn 3774 3011 a(N) p Ff 3862 3011 a(\)) 0 3131 y(such) p 222 3131 a(that) p Fk 1483 3252 a(jj) p Fn(E) p Fo 1617 3252 a(\() p Fq(c) p Fo(\)) p Fk(jj) p Fl 1799 3210 a(2) p Fm 1799 3276 a(N) p Fk 1893 3252 a(\024) p Fn 1998 3252 a(C) p Fl 2068 3267 a(3) p Fk 2107 3252 a(jj) p Fq(c) p Fk(jj) p Fl 2269 3210 a(2) p Fa 2269 3277 a(C) p Fd 2328 3258 a(4) p Fe(N) p Fo 3653 3252 a(\(5.10\)) p Ff 0 3413 a(for) p 156 3413 a(any) p Fq 342 3413 a(c) p Fk 420 3413 a(2) p Fj 514 3413 a(C) p Fl 602 3371 a(4) p Fm(N) p Ff 705 3413 a(.) 146 3534 y(\(iii\)) p Fn 351 3534 a(u) p Fk 428 3534 a(\000) p Fn 528 3534 a(E) p Fo 606 3534 a(\(\004) p Fn(u) p Fo(\)) p Fk 830 3534 a(2) p Fn 924 3534 a(D) p Fo 1008 3534 a(\() p 1046 3456 134 4 v Fn(L) p Fm 1112 3549 a(N) p Fo 1179 3534 a(\)) p Ff 1252 3534 a(for) p 1408 3534 a(every) p Fn 1666 3534 a(u) p Fk 1749 3534 a(2) p Fn 1843 3534 a(D) p Fo 1927 3534 a(\() p Fn(L) p Fg 2031 3497 a(\003) p Fm 2031 3558 a(N) p Fo 2098 3534 a(\)) p Ff(.) 146 3654 y(\(iv\)) p Fo 336 3654 a(\004) p Fn(E) p Fo 479 3654 a(\() p Fq(c) p Fo(\)) p 632 3654 a(=) p Fq 736 3654 a(c) p Ff 820 3654 a(for) p 976 3654 a(every) p Fq 1234 3654 a(c) p Fk 1311 3654 a(2) p Fj 1405 3654 a(C) p Fl 1494 3612 a(4) p Fm(N) p Ff 1596 3654 a(.) 146 3774 y(\(v\)) p Fo 306 3774 a(\004) p Fn(u) p Fo 454 3774 a(=) p 558 3774 a(0) p Ff 641 3774 a(for) p 797 3774 a(every) p 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2794 4089 a(w) m(e) p 2937 4089 a(ha) m(v) m(e) p 3162 4089 a(that) p 3373 4089 a(there) p 3623 4089 a(exist) p 3853 4089 a(a) 0 4210 y(constan) m(t) p Fn 393 4210 a(C) p Fo 503 4210 a(dep) s(enden) m(t) p 974 4210 a(only) p 1188 4210 a(on) p Fn 1324 4210 a(B) p Fo 1403 4210 a(,) p Fn 1463 4210 a(\013) p Fg 1525 4225 a(\000) p Fo 1584 4210 a(,) p Fn 1643 4210 a(\013) p Fl 1705 4225 a(+) p Fo 1797 4210 a(suc) m(h) p 2017 4210 a(that) 983 4400 y(sup) p Fm 882 4478 a(\013) p Fb 927 4487 a(\000) p Fg 980 4478 a(\024) p Fm(\013) p Fg(\024) p Fm(\013) p Fd 1180 4487 a(+) p Fo 1248 4400 a(max) p Fk 1429 4400 a(fjj) p Fn(\036) p Fm 1593 4359 a(\013) p Fg 1593 4425 a(\000) p Fl(1) p Fk 1686 4400 a(jj) p Fl 1742 4359 a(2) 1742 4425 y(1) p Fn 1781 4400 a(;) p Fk 1825 4400 a(jj) p Fn( ) p Fm 1948 4359 a(\013) p Fl 1944 4425 a(1) p Fk 1997 4400 a(jj) p Fl 2053 4359 a(2) 2053 4425 y(1) p Fn 2091 4400 a(;) p Fk 2135 4400 a(jj) p Fn(\036) p Fm 2249 4359 a(\013) p Fl 2249 4425 a(0) p Fk 2298 4400 a(jj) p Fl 2354 4359 a(2) 2354 4425 y(1) p Fn 2392 4400 a(;) p Fk 2436 4400 a(jj) p Fn(\036) p Fm 2550 4359 a(\013) p Fg 2550 4425 a(\000) p Fl(1) p Fk 2644 4400 a(jj) p Fl 2700 4359 a(2) 2700 4425 y(1) p Fk 2738 4400 a(g) p 2816 4400 a(\024) p Fn 2921 4400 a(C) r(:) p Fo 3653 4400 a(\(5.11\)) 0 4655 y(By) p 152 4655 a(\(iii\)) p 341 4655 a(of) p 452 4655 a(Lemma) p 799 4655 a(3.3,) p 983 4655 a(w) m(e) p 1125 4655 a(ha) m(v) m(e) p 1350 4655 a(that) p 1560 4655 a(there) p 1809 4655 a(exist) p 2039 4655 a(a) p 2119 4655 a(constan) m(t) p Fn 2512 4655 a(C) p Fg 2589 4619 a(0) p Fo 2644 4655 a(dep) s(enden) m(t) p 3115 4655 a(only) p 3328 4655 a(on) p Fn 3463 4655 a(B) p Fo 3542 4655 a(,) p Fn 3601 4655 a(\013) p Fg 3663 4670 a(\000) p Fo 3722 4655 a(,) p Fn 3781 4655 a(\013) p Fl 3843 4670 a(+) p Fo 0 4776 a(suc) m(h) p 220 4776 a(that) 965 4896 y(sup) p Fm 864 4974 a(\013) p Fb 909 4983 a(\000) p Fg 962 4974 a(\024) p Fm(\013) p Fg(\024) p Fm(\013) p Fd 1162 4983 a(+) p Fo 1229 4896 a(max) p Fk(fjj) p Fo(\010) p Fm 1587 4855 a(\013) p Fg 1587 4921 a(\000) p Fl(1) p Fk 1681 4896 a(jj) p Fl 1737 4855 a(2) p Fn 1776 4896 a(;) p Fk 1820 4896 a(jj) p Fo(\011) p Fm 1952 4855 a(\013) p Fl 1952 4921 a(1) p Fk 2000 4896 a(jj) p Fl 2056 4855 a(2) p Fn 2095 4896 a(;) p Fk 2139 4896 a(jj) p Fo(\010) p Fm 2265 4855 a(\013) p Fl 2265 4921 a(0) p Fk 2314 4896 a(jj) p Fl 2370 4855 a(2) p Fn 2409 4896 a(;) p Fk 2453 4896 a(jj) p Fo(\011) p Fm 2585 4855 a(\013) p Fl 2585 4921 a(0) p Fk 2633 4896 a(jj) p Fl 2689 4855 a(2) p Fk 2728 4896 a(g) p 2805 4896 a(\024) p Fn 2910 4896 a(C) p Fg 2987 4855 a(0) p Fn 3011 4896 a(:) p Fo 3653 4896 a(\(5.12\)) 1902 5214 y(29) p eop %%Page: 30 30 30 29 bop Fo 0 631 a(By) p 153 631 a(\(5.11\),) p 462 631 a(\(5.12\)) p 744 631 a(and) p 933 631 a(\(4.15\),) p 1242 631 a(w) m(e) p 1386 631 a(ha) m(v) m(e) p Fm 867 778 a(N) p Fi 838 803 a(X) p Fm 837 986 a(j) p Fl 870 986 a(=1) p Fo 960 886 a(\() p Fk(j) p Fo(\010) p Fl 1096 836 a(\() p Fm(j) p Fl 1156 836 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1770 a(1) p Fk 1095 1755 a(jj) p Fn(u) p Fk(jj) p Fl 1263 1714 a(2) p Fm 1263 1779 a(N) p Fn 1329 1755 a(:) p Fo 0 1936 a(Th) m(us) p 247 1936 a(\004) p Fn(u) p Fk 395 1936 a(2) p Fj 489 1936 a(C) p Fl 578 1894 a(4) p Fm(N) p Fo 713 1936 a(and) p 903 1936 a(\(5.8\)) p 1136 1936 a(holds) p 1391 1936 a(for) p Fn 1540 1936 a(C) p Fl 1610 1951 a(2) p Fo 1677 1936 a(=) p 1780 1936 a(4) p Fn(C) p Fg 1906 1899 a(0) p Fn 1929 1936 a(C) p Fl 1999 1951 a(1) p Fo 2039 1936 a(.) 146 2056 y(\(ii\)) p 320 2056 a(Notice) p 638 2056 a(that) p 861 2056 a(t) m(w) m(o) p 1057 2056 a(basis) p 1308 2056 a(functions) p 1740 2056 a(are) p 1914 2056 a(m) m(utually) p 2337 2056 a(orthogonal) p 2837 2056 a(if) p 2937 2056 a(the) p 3117 2056 a(upp) s(er) p 3408 2056 a(index) p Fn 3682 2056 a(j) p Fo 3772 2056 a(are) 0 2176 y(di\013eren) m(t.) p 423 2176 a(F) p 479 2176 a(rom) p 679 2176 a(this) p 869 2176 a(fact,) p 1088 2176 a(\(5.11\)) p 1370 2176 a(and) p 1560 2176 a(\(4.6\),) p 1820 2176 a(w) m(e) p 1964 2176 a(ha) m(v) m(e) p 2189 2176 a(for) p Fq 2338 2176 a(c) p Fk 2415 2176 a(2) p Fj 2509 2176 a(C) p Fl 2598 2134 a(4) p Fm(N) p Fk 1026 2434 a(jj) p Fm 1142 2324 a(N) p Fd 1198 2333 a(2) p Fi 1127 2351 a(X) p Fm 1098 2534 a(j) p Fl 1131 2534 a(=) p Fm(N) p Fd 1242 2543 a(1) p Fn 1293 2434 a(c) p Fl 1335 2449 a(4) p Fm(j) p Fg 1403 2449 a(\000) p Fl(3) p Fn 1497 2434 a(\036) p Fl 1555 2383 a(\() p Fm(j) p Fl 1615 2383 a(\)) p Fg 1555 2455 a(\000) p Fl(1) p Fo 1671 2434 a(+) p Fn 1769 2434 a(c) p Fl 1811 2449 a(4) p Fm(j) p Fg 1879 2449 a(\000) p Fl(2) p Fn 1973 2434 a( ) p Fl 2040 2383 a(\() p Fm(j) p Fl 2100 2383 a(\)) 2036 2455 y(1) p Fo 2153 2434 a(+) p Fn 2251 2434 a(c) p Fl 2293 2449 a(4) p Fm(j) p Fg 2361 2449 a(\000) p Fl(1) p Fn 2455 2434 a(\036) p Fl 2513 2383 a(\() p Fm(j) p Fl 2573 2383 a(\)) 2513 2455 y(0) p Fo 2627 2434 a(+) p Fn 2725 2434 a(c) p Fl 2767 2449 a(4) p Fm(j) p Fn 2838 2434 a( ) p Fl 2905 2383 a(\() p Fm(j) p Fl 2965 2383 a(\)) 2901 2455 y(0) p Fk 2997 2434 a(jj) p Fl 3053 2393 a(2) p Fm 3053 2458 a(N) p Fk 866 2752 a(\024) p Fm 1070 2643 a(N) p Fd 1126 2652 a(2) p Fi 1055 2669 a(X) p Fm 1026 2853 a(j) p Fl 1059 2853 a(=) p Fm(N) p Fd 1170 2862 a(1) p Fk 1221 2752 a(jj) p Fn(c) p Fl 1319 2767 a(4) p Fm(j) p Fg 1387 2767 a(\000) p Fl(3) p Fn 1480 2752 a(\036) p Fl 1538 2701 a(\() p Fm(j) p Fl 1598 2701 a(\)) p Fg 1538 2774 a(\000) p Fl(1) p Fo 1654 2752 a(+) p Fn 1752 2752 a(c) p Fl 1794 2767 a(4) p Fm(j) p Fg 1862 2767 a(\000) p Fl(2) p Fn 1956 2752 a( ) p Fl 2023 2701 a(\() p Fm(j) p Fl 2083 2701 a(\)) 2019 2774 y(1) p Fo 2137 2752 a(+) p Fn 2235 2752 a(c) p Fl 2277 2767 a(4) p Fm(j) p Fg 2345 2767 a(\000) p Fl(1) p Fn 2439 2752 a(\036) p Fl 2497 2701 a(\() p Fm(j) p Fl 2557 2701 a(\)) 2497 2774 y(0) p Fo 2610 2752 a(+) p Fn 2708 2752 a(c) p Fl 2750 2767 a(4) p Fm(j) p Fn 2822 2752 a( ) p Fl 2889 2701 a(\() p Fm(j) p Fl 2949 2701 a(\)) 2885 2774 y(0) p Fk 2980 2752 a(jj) p Fl 3036 2711 a(2) p Fm 3036 2777 a(N) p Fk 866 3071 a(\024) p Fo 1026 3071 a(4) p Fn(C) p 1152 3071 a(C) p Fl 1222 3086 a(1) p Fm 1321 2961 a(N) p Fd 1377 2970 a(2) p Fi 1307 2988 a(X) p Fm 1278 3172 a(j) p Fl 1311 3172 a(=) p Fm(N) p Fd 1422 3181 a(1) p Fo 1456 3071 a(\() p Fk(j) p Fn(c) p Fl 1564 3086 a(4) p Fm(j) p Fg 1632 3086 a(\000) p Fl(3) p Fk 1725 3071 a(j) p Fl 1753 3030 a(2) p Fo 1814 3071 a(+) p Fk 1912 3071 a(j) p Fn(c) p Fl 1982 3086 a(4) p Fm(j) p Fg 2050 3086 a(\000) p Fl(2) p Fk 2144 3071 a(j) p Fl 2172 3030 a(2) p Fo 2233 3071 a(+) p Fk 2331 3071 a(j) p Fn(c) p Fl 2401 3086 a(4) p Fm(j) p Fg 2469 3086 a(\000) p Fl(1) p Fk 2563 3071 a(j) p Fl 2591 3030 a(2) p Fo 2652 3071 a(+) p Fk 2750 3071 a(j) p Fn(c) p Fl 2820 3086 a(4) p Fm(j) p Fk 2892 3071 a(j) p Fl 2920 3030 a(2) p Fo 2959 3071 a(\)) p Fn(;) p Fo 3653 3071 a(\(5.13\)) 0 3331 y(for) p 153 3331 a(an) m(y) p Fn 341 3331 a(N) p Fl 419 3346 a(1) p Fo 459 3331 a(,) p Fn 523 3331 a(N) p Fl 601 3346 a(2) p Fo 677 3331 a(suc) m(h) p 901 3331 a(that) p 1117 3331 a(1) p Fk 1200 3331 a(\024) p Fn 1312 3331 a(N) p Fl 1390 3346 a(1) p Fk 1464 3331 a(\024) p Fn 1575 3331 a(N) p Fl 1653 3346 a(2) p Fk 1727 3331 a(\024) p Fn 1839 3331 a(N) p Fo 1964 3331 a(and) p Fn 2158 3331 a(N) p Fl 2236 3346 a(2) p Fn 2310 3331 a(<) p Fk 2420 3331 a(1) p Fo(.) p 2602 3331 a(When) p Fn 2890 3331 a(N) p Fo 3013 3331 a(=) p Fk 3124 3331 a(1) p Fo(,) p 3288 3331 a(this) p 3482 3331 a(inequalit) m(y) 0 3451 y(guaran) m(tees) p 499 3451 a(the) p 680 3451 a(con) m(v) m (ergence) p 1233 3451 a(of) p 1357 3451 a(the) p 1539 3451 a(in\014nite) p 1883 3451 a(sum) p 2102 3451 a(\(5.9\),) p 2379 3451 a(since) p 2632 3451 a(the) p 2813 3451 a(righ) m(t) p 3062 3451 a(hand) p 3319 3451 a(side) p 3528 3451 a(of) p 3653 3451 a(\(5.13\)) 0 3572 y(con) m(v) m(erges) p 433 3572 a(to) p 549 3572 a(0) p 627 3572 a(as) p Fn 744 3572 a(N) p Fl 822 3587 a(1) p Fn 861 3572 a(;) p 905 3572 a(N) p Fl 983 3587 a(2) p Fk 1050 3572 a(!) p 1178 3572 a(1) p Fo(.) p 1347 3572 a(Moreo) m(v) m(er,) p 1799 3572 a(taking) p Fn 2097 3572 a(N) p Fl 2175 3587 a(1) p Fo 2242 3572 a(=) p 2345 3572 a(1) p 2423 3572 a(and) p Fn 2610 3572 a(N) p Fl 2688 3587 a(2) p Fo 2755 3572 a(=) p Fn 2858 3572 a(N) p Fo 2976 3572 a(\(or) p Fn 3130 3572 a(N) p Fl 3208 3587 a(2) p Fk 3275 3572 a(!) p 3403 3572 a(1) p Fo(\)) p 3569 3572 a(w) m(e) p 3710 3572 a(ha) m(v) m(e) 0 3692 y(that) p 211 3692 a(\(5.10\)) p 493 3692 a(holds) p 748 3692 a(with) p Fn 971 3692 a(C) p Fl 1041 3707 a(3) p Fo 1108 3692 a(=) p 1211 3692 a(4) p Fn(C) p 1337 3692 a(C) p Fl 1407 3707 a(1) p Fo 1446 3692 a(.) 146 3813 y(\(iii\)) p 335 3813 a(By) p 486 3813 a(\(v\)) p 644 3813 a(of) p 754 3813 a(Lemma) p 1101 3813 a(4.4) p 1256 3813 a(and) p 1445 3813 a(\(iv\)) p 1630 3813 a(of) p 1740 3813 a(Prop) s(osition) p 2263 3813 a(3.2,) p 2446 3813 a(it) p 2542 3813 a(su\016ces) p 2872 3813 a(to) p 2990 3813 a(sho) m(w) p Fn 3230 3813 a(u) p Fl 3286 3828 a(0) p Fo 3353 3813 a(=) p Fn 3456 3813 a(u) p Fk 3531 3813 a(\000) p Fn 3628 3813 a(E) p Fo 3706 3813 a(\(\004) p Fn(u) p Fo(\)) 0 3933 y(satis\014es) 992 4053 y(\010) p Fl 1062 4002 a(\() p Fm(j) p Fl 1122 4002 a(\)) p Fg 1062 4075 a(\000) p Fl(1) p Fo 1157 4053 a(\() p Fn(u) p Fl 1251 4068 a(0) p Fo 1290 4053 a(\)) p 1355 4053 a(=) p 1459 4053 a(\011) p Fl 1535 4002 a(\() p Fm(j) p Fl 1595 4002 a(\)) 1535 4075 y(1) p Fo 1626 4053 a(\() p Fn(u) p Fl 1720 4068 a(0) p Fo 1759 4053 a(\)) p 1825 4053 a(=) p 1928 4053 a(\010) p Fl 1998 4002 a(\() p Fm(j) p Fl 2058 4002 a(\)) 1998 4075 y(0) p Fo 2090 4053 a(\() p Fn(u) p Fl 2184 4068 a(0) p Fo 2223 4053 a(\)) p 2288 4053 a(=) p 2392 4053 a(\011) p Fl 2468 4002 a(\() p Fm(j) p Fl 2528 4002 a(\)) 2468 4075 y(0) p Fo 2559 4053 a(\() p Fn(u) p Fl 2653 4068 a(0) p Fo 2692 4053 a(\)) p 2758 4053 a(=) p 2861 4053 a(0) 0 4211 y(for) p 142 4211 a(ev) m(ery) p Fn 392 4211 a(j) p Fo 466 4211 a(=) p 570 4211 a(1) p Fn(;) p 663 4211 a(:) p 707 4211 a(:) p 751 4211 a(:) p 793 4211 a(;) p 837 4211 a(N) p Fo 925 4211 a(.) p 994 4211 a(These) p 1269 4211 a(equalities) p 1693 4211 a(follo) m(ws) p 2007 4211 a(from) p 2230 4211 a(the) p 2391 4211 a(de\014nition) p 2817 4211 a(of) p Fn 2921 4211 a(E) p Fo 2999 4211 a(,) p 3053 4211 a(\004) p 3144 4211 a(and) p 3327 4211 a(\(ii\)) p 3482 4211 a(of) p 3586 4211 a(Lemma) 0 4331 y(5.1.) 146 4452 y(\(iv\)\(v\)) p 461 4452 a(These) p 743 4452 a(assertions) p 1189 4452 a(follo) m(w) p 1471 4452 a(from) p 1701 4452 a(the) p 1869 4452 a(de\014nition) p 2303 4452 a(of) p Fn 2414 4452 a(E) p Fo 2492 4452 a(,) p 2551 4452 a(\004) p 2649 4452 a(and) p 2839 4452 a(\(ii\)) p 3001 4452 a(of) p 3112 4452 a(Lemma) p 3461 4452 a(5.1.) p 3656 4384 64 4 v 3656 4477 4 93 v 3716 4477 V 3656 4480 64 4 v 146 4635 a(In) p 266 4635 a(the) p 431 4635 a(sequel,) p 746 4635 a(w) m(e) p 887 4635 a(denote) p 1199 4635 a(the) p 1364 4635 a(algebraic) p 1773 4635 a(direct) p 2047 4635 a(sum) p 2251 4635 a(of) p 2359 4635 a(t) m(w) m(o) p 2540 4635 a(subspace) p Fn 2945 4635 a(W) p Fl 3037 4650 a(1) p Fo 3107 4635 a(and) p Fn 3294 4635 a(W) p Fl 3386 4650 a(2) p Fo 3455 4635 a(of) p 3563 4635 a(a) p 3642 4635 a(v) m(ector) 0 4756 y(space) p Fn 261 4756 a(V) p Fo 372 4756 a(b) m(y) p Fn 507 4756 a(W) p Fl 599 4771 a(1) p Fk 661 4756 a(\010) p Fm 738 4771 a(al) q(g) p Fn 860 4756 a(W) p Fl 952 4771 a(2) p Fo 992 4756 a(,) p 1051 4756 a(that) p 1262 4756 a(is,) p Fn 856 4937 a(W) p Fl 948 4952 a(1) p Fk 1009 4937 a(\010) p Fm 1086 4952 a(al) q(g) p Fn 1208 4937 a(W) p Fl 1300 4952 a(2) p Fo 1367 4937 a(=) p Fk 1471 4937 a(f) p Fn(w) p Fl 1591 4952 a(1) p Fo 1652 4937 a(+) p Fn 1750 4937 a(w) p Fl 1820 4952 a(2) p Fk 1887 4937 a(2) p Fn 1981 4937 a(V) p Fo 2092 4937 a(;) p Fn 2168 4937 a(w) p Fl 2238 4952 a(1) p Fk 2305 4937 a(2) p Fn 2399 4937 a(W) p Fl 2491 4952 a(1) p Fn 2531 4937 a(;) p 2607 4937 a(w) p Fl 2677 4952 a(2) p Fk 2744 4937 a(2) p Fn 2838 4937 a(W) p Fl 2930 4952 a(2) p Fk 2970 4937 a(g) p Fn(:) p Fo 1902 5214 a(30) p eop %%Page: 31 31 31 30 bop Fq 146 631 a(Prop) s(osition) p 752 631 a(5.3) p Ff 944 631 a(L) p 1000 631 a(et) p Fo 1112 631 a(1) p Fk 1189 631 a(\024) p Fn 1295 631 a(N) p 1411 631 a(<) p Fk 1515 631 a(1) p Ff 1650 631 a(or) p Fn 1776 631 a(N) p Fo 1893 631 a(=) p Fk 1997 631 a(1) p Ff(.) p 2172 631 a(Supp) p 2374 631 a(ose) p 2543 631 a(that) p 2742 631 a(Assumption) p 3278 631 a(1) p 3363 631 a(holds.) p 3653 631 a(Then,) 0 751 y(the) p 162 751 a(op) p 257 751 a(er) p 338 751 a(ator) p 545 751 a(domain) p Fn 894 751 a(D) p Fo 978 751 a(\() p Fn(L) p Fg 1082 715 a(\003) p Fm 1082 776 a(N) p Fo 1149 751 a(\)) p Ff 1222 751 a(is) p 1327 751 a(de) p 1417 751 a(c) p 1457 751 a(omp) p 1632 751 a(ose) p 1762 751 a(d) p 1845 751 a(into) p 2047 751 a(a) p 2131 751 a(dir) p 2247 751 a(e) p 2287 751 a(ct) p 2399 751 a(sum) p Fn 889 974 a(D) p Fo 973 974 a(\() p Fn(L) p Fg 1077 933 a(\003) p Fm 1077 998 a(N) p Fo 1145 974 a(\)) p 1210 974 a(=) p Fn 1314 974 a(D) p Fo 1398 974 a(\() p 1436 896 134 4 v Fn(L) p Fm 1502 989 a(N) p Fo 1569 974 a(\)) p Fk 1629 974 a(\010) p Fm 1706 989 a(al) q(g) 1858 899 y(N) p Fk 1851 974 a(\010) p Fm 1828 1040 a(j) p Fl 1861 1040 a(=1) p Fo 1967 974 a(L) p Fn(:) p Fo(h) p Fn(:) p Fk 2137 974 a(f) p Fn(\036) p Fl 2245 923 a(\() p Fm(j) p Fl 2305 923 a(\)) p Fg 2245 995 a(\000) p Fl(1) p Fn 2339 974 a(;) p 2383 974 a( ) p Fl 2450 923 a(\() p Fm(j) p Fl 2510 923 a(\)) 2446 995 y(1) p Fn 2541 974 a(;) p 2585 974 a(\036) p Fl 2643 923 a(\() p Fm(j) p Fl 2703 923 a(\)) 2643 995 y(0) p Fn 2734 974 a(;) p 2778 974 a( ) p Fl 2845 923 a(\() p Fm(j) p Fl 2905 923 a(\)) 2841 995 y(0) p Fk 2936 974 a(g) p Fn(;) p Ff 0 1211 a(wher) p 196 1211 a(e) p Fk 279 1211 a(\010) p Fm 356 1226 a(al) q(g) p Ff 494 1211 a(denotes) p 849 1211 a(the) p 1014 1211 a(algebr) p 1260 1211 a(aic) p 1423 1211 a(dir) p 1539 1211 a(e) p 1579 1211 a(ct) p 1694 1211 a(sum) p 1904 1211 a(and) p Fk 2097 1211 a(\010) p Fm 2175 1175 a(N) 2175 1236 y(j) p Fl 2208 1236 a(=1) p Ff 2340 1211 a(denotes) p 2695 1211 a(the) p 2860 1211 a(ortho) p 3078 1211 a(gonal) p 3341 1211 a(dir) p 3457 1211 a(e) p 3497 1211 a(ct) p 3612 1211 a(sum) p 3822 1211 a(of) 0 1332 y(mutual) p 296 1332 a(ly) p 403 1332 a(ortho) p 621 1332 a(gonal) p 880 1332 a(close) p 1080 1332 a(d) p 1164 1332 a(subsp) p 1386 1332 a(ac) p 1476 1332 a(es.) p 1635 1332 a(Mor) p 1808 1332 a(e) p 1888 1332 a(pr) p 1974 1332 a(e) p 2014 1332 a(cisely,) p 2310 1332 a(every) p Fn 2568 1332 a(u) p Fk 2651 1332 a(2) p Fn 2745 1332 a(D) p Fo 2829 1332 a(\() p Fn(L) p Fg 2933 1296 a(\003) p Fm 2933 1356 a(N) p Fo 3001 1332 a(\)) p Ff 3073 1332 a(is) p 3178 1332 a(de) p 3268 1332 a(c) p 3308 1332 a(omp) p 3483 1332 a(ose) p 3613 1332 a(d) p 3696 1332 a(as) p Fn 476 1513 a(u) p Fo 615 1513 a(=) p Fn 774 1513 a(u) p Fl 830 1528 a(0) p Fo 891 1513 a(+) p Fn 989 1513 a(E) p Fo 1067 1513 a(\(\004) p Fn(u) p Fo(\)) 615 1733 y(=) p Fn 774 1733 a(u) p Fl 830 1748 a(0) p Fo 891 1733 a(+) p Fm 1018 1625 a(N) p Fi 990 1650 a(X) p Fm 989 1832 a(j) p Fl 1022 1832 a(=1) p Fi 1111 1611 a(\022) p Fo 1172 1733 a(\010) p Fl 1242 1682 a(\() p Fm(j) p Fl 1302 1682 a(\)) p Fg 1242 1754 a(\000) p Fl(1) p Fo 1337 1733 a(\() p Fn(u) p Fo(\)) p Fn(\036) p Fl 1527 1682 a(\() p Fm(j) p Fl 1587 1682 a(\)) p Fg 1527 1754 a(\000) p Fl(1) p Fo 1643 1733 a(+) p 1741 1733 a(\011) p Fl 1817 1682 a(\() p Fm(j) p Fl 1877 1682 a(\)) 1817 1754 y(1) p Fo 1908 1733 a(\() p Fn(u) p Fo(\)) p Fn( ) p Fl 2107 1682 a(\() p Fm(j) p Fl 2167 1682 a(\)) 2103 1754 y(1) p Fo 2220 1733 a(+) p 2318 1733 a(\010) p Fl 2388 1682 a(\() p Fm(j) p Fl 2448 1682 a(\)) 2388 1754 y(0) p Fo 2480 1733 a(\() p Fn(u) p Fo(\)) p Fn(\036) p Fl 2670 1682 a(\() p Fm(j) p Fl 2730 1682 a(\)) 2670 1754 y(0) p Fo 2783 1733 a(+) p 2881 1733 a(\011) p Fl 2957 1682 a(\() p Fm(j) p Fl 3017 1682 a(\)) 2957 1754 y(0) p Fo 3048 1733 a(\() p Fn(u) p Fo(\)) p Fn( ) p Fl 3247 1682 a(\() p Fm(j) p Fl 3307 1682 a(\)) 3243 1754 y(0) p Fi 3338 1611 a(\023) p Fn 3399 1733 a(;) p Fo 3653 1733 a(\(5.14\)) p Ff 0 2007 a(wher) p 196 2007 a(e) p Fn 282 2007 a(u) p Fl 338 2022 a(0) p Fk 416 2007 a(2) p Fn 523 2007 a(D) p Fo 607 2007 a(\() p 645 1929 V Fn(L) p Fm 711 2022 a(N) p Fo 778 2007 a(\)) p Ff 857 2007 a(\(when) p Fn 1153 2007 a(N) p Fo 1281 2007 a(=) p Fk 1397 2007 a(1) p Ff(,) p 1569 2007 a(the) p 1738 2007 a(right) p 1977 2007 a(hand) p 2223 2007 a(side) p 2428 2007 a(of) p 2550 2007 a(\(5.14\)) p 2850 2007 a(is) p 2961 2007 a(a) p 3052 2007 a(c) p 3092 2007 a(onver) p 3323 2007 a(gent) p 3541 2007 a(se) p 3621 2007 a(quenc) p 3858 2007 a(e) 0 2127 y(in) p Fn 121 2127 a(D) p Fo 205 2127 a(\() p Fn(L) p Fg 309 2091 a(\003) p Fm 309 2152 a(N) p Fo 376 2127 a(\)) p Ff(\).) p 534 2127 a(This) p 760 2127 a(de) p 850 2127 a(c) p 890 2127 a(omp) p 1065 2127 a(osition) p 1386 2127 a(is) p 1493 2127 a(unique) p 1808 2127 a(in) p 1929 2127 a(the) p 2093 2127 a(sense) p 2353 2127 a(that) p 2554 2127 a(if) p Fn 2651 2127 a(u) p Fl 2707 2142 a(0) p Fo 2769 2127 a(+) p Fn 2868 2127 a(u) p Fl 2924 2142 a(1) p Ff 2999 2127 a(=) p Fn 3111 2127 a(u) p Fg 3167 2091 a(0) p Fl 3167 2152 a(0) p Fo 3229 2127 a(+) p Fn 3328 2127 a(u) p Fg 3384 2091 a(0) p Fl 3384 2152 a(1) p Ff 3423 2127 a(,) p 3490 2127 a(and) p 3681 2127 a(if) p Fn 3777 2127 a(u) p Fl 3833 2142 a(0) p Ff 3872 2127 a(,) p Fn 0 2247 a(u) p Fg 56 2211 a(0) p Fl 56 2272 a(0) p Fk 123 2247 a(2) p Fn 217 2247 a(D) p Fo 301 2247 a(\() p 339 2169 V Fn(L) p Fm 405 2262 a(N) p Fo 472 2247 a(\)) p Ff 545 2247 a(and) p Fn 734 2247 a(u) p Fl 790 2262 a(1) p Ff 829 2247 a(,) p Fn 894 2247 a(u) p Fg 950 2211 a(0) p Fl 950 2272 a(1) p Fk 1016 2247 a(2) p Fo 1111 2247 a(Ran) p Fn 1302 2247 a(E) p Ff 1380 2247 a(,) p 1445 2247 a(then) p 1661 2247 a(we) p 1806 2247 a(have) p Fn 2030 2247 a(u) p Fl 2086 2262 a(0) p Fo 2153 2247 a(=) p Fn 2256 2247 a(u) p Fg 2312 2211 a(0) p Fl 2312 2272 a(0) p Ff 2386 2247 a(and) p Fn 2575 2247 a(u) p Fl 2631 2262 a(1) p Fo 2698 2247 a(=) p Fn 2801 2247 a(u) p Fg 2857 2211 a(0) p Fl 2857 2272 a(1) p Ff 2896 2247 a(.) p 2971 2247 a(The) p 3170 2247 a(map) p Fi 1043 2396 a(e) p Fo 1033 2429 a(\004) p 1126 2429 a(:) p Fn 1181 2429 a(D) p Fo 1265 2429 a(\() p Fn(L) p Fg 1369 2387 a(\003) p Fm 1369 2453 a(N) p Fo 1436 2429 a(\)) p Fk 1502 2429 a(3) p Fn 1596 2429 a(u) p Fk 1679 2429 a(7!) p Fo 1806 2429 a(\() p Fn(u) p Fl 1900 2444 a(0) p Fn 1939 2429 a(;) p Fo 1983 2429 a(\004) p Fn(u) p Fo(\)) p Fk 2169 2429 a(2) p Fn 2263 2429 a(D) p Fo 2347 2429 a(\() p 2385 2351 V Fn(L) p Fm 2451 2444 a(N) p Fo 2518 2429 a(\)) p Fk 2578 2429 a(\010) p Fj 2678 2429 a(C) p Fl 2766 2386 a(4) p Fm(N) p Fo 3653 2429 a(\(5.15\)) p Ff 0 2610 a(is) p 103 2610 a(a) p 186 2610 a(Banach) p 536 2610 a(sp) p 621 2610 a(ac) p 711 2610 a(e) p 789 2610 a(isomorphism,) p 1391 2610 a(wher) p 1587 2610 a(e) p Fn 1664 2610 a(D) p Fo 1748 2610 a(\() p 1786 2532 V Fn(L) p Fm 1852 2625 a(N) p Fo 1920 2610 a(\)) p Fk 1976 2610 a(\010) p Fj 2071 2610 a(C) p Fl 2160 2568 a(4) p Fm(N) p Ff 2295 2610 a(is) p 2398 2610 a(the) p 2558 2610 a(ortho) p 2776 2610 a(gonal) p 3033 2610 a(dir) p 3149 2610 a(e) p 3189 2610 a(ct) p 3299 2610 a(sum) p 3504 2610 a(of) p 3617 2610 a(Hilb) p 3784 2610 a(ert) 0 2730 y(sp) p 85 2730 a(ac) p 175 2730 a(es.) p 334 2730 a(The) p 533 2730 a(inverse) p 868 2730 a(map) p 1082 2730 a(of) p Fi 1206 2698 a(e) p Fo 1197 2730 a(\004) p Ff 1297 2730 a(is) p 1401 2730 a(given) p 1655 2730 a(by) p Fi 1081 2879 a(e) p Fo 1072 2911 a(\004) p Fg 1137 2870 a(\000) p Fl(1) p Fo 1259 2911 a(:) p Fn 1313 2911 a(D) p Fo 1397 2911 a(\() p 1435 2833 V Fn(L) p Fm 1501 2926 a(N) p Fo 1569 2911 a(\)) p Fk 1629 2911 a(\010) p Fj 1728 2911 a(C) p Fl 1817 2869 a(4) p Fm(N) p Fk 1947 2911 a(3) p Fo 2041 2911 a(\() p Fn(u) p Fl 2135 2926 a(0) p Fn 2174 2911 a(;) p Fq 2218 2911 a(c) p Fo(\)) p Fk 2333 2911 a(7!) p Fn 2460 2911 a(u) p Fl 2516 2926 a(0) p Fo 2577 2911 a(+) p Fn 2675 2911 a(E) p Fq 2753 2911 a(c) p Fn(:) p Fo 3653 2911 a(\(5.16\)) p Ff 0 3092 a(Mor) p 173 3092 a(e) p 213 3092 a(over,) p 467 3092 a(ther) p 630 3092 a(e) p 717 3092 a(exist) p 951 3092 a(a) p 1043 3092 a(c) p 1083 3092 a(onstant) p Fn 1438 3092 a(C) p Fl 1508 3107 a(4) p Fn 1588 3092 a(>) p Fo 1705 3092 a(1) p Ff 1796 3092 a(dep) p 1936 3092 a(endent) p 2259 3092 a(on) p Fn 2405 3092 a(B) p Ff 2484 3092 a(,) p Fn 2558 3092 a(R) p Ff 2633 3092 a(,) p Fn 2707 3092 a(\013) p Fg 2769 3107 a(\000) p Ff 2828 3092 a(,) p Fn 2901 3092 a(\013) p Fl 2963 3107 a(+) p Ff 3065 3092 a(\(indep) p 3330 3092 a(endent) p 3652 3092 a(of) p Fn 3774 3092 a(N) p Ff 3862 3092 a(\)) 0 3213 y(such) p 222 3213 a(that) p Fn 1057 3333 a(C) p Fg 1134 3292 a(\000) p Fl(1) 1127 3358 y(4) p Fk 1228 3333 a(jj) p Fn(u) p Fk(jj) p Fl 1396 3292 a(2) p Fm 1396 3358 a(N) p Fk 1489 3333 a(\024) p 1594 3333 a(jj) p Fn(u) p Fl 1706 3348 a(0) p Fk 1745 3333 a(jj) p Fl 1801 3292 a(2) p Fm 1801 3358 a(N) p Fo 1889 3333 a(+) p Fk 1987 3333 a(jj) p Fo(\004) p Fn(u) p Fk(jj) p Fl 2220 3292 a(2) p Fa 2220 3358 a(C) p Fd 2279 3339 a(4) p Fe(N) p Fk 2397 3333 a(\024) p Fn 2503 3333 a(C) p Fl 2573 3348 a(4) p Fk 2612 3333 a(jj) p Fn(u) p Fk(jj) p Fl 2780 3292 a(2) p Fm 2780 3358 a(N) p Fo 3653 3333 a(\(5.17\)) p Ff 0 3491 a(for) p 156 3491 a(every) p Fn 413 3491 a(u) p Fk 497 3491 a(2) p Fn 591 3491 a(D) p Fo 675 3491 a(\() p Fn(L) p Fg 779 3454 a(\003) p Fm 779 3515 a(N) p Fo 846 3491 a(\)) p Ff(.) 146 3674 y(Pr) p 248 3674 a(o) p 293 3674 a(of.) p Fo 427 3674 a(W) p 519 3674 a(e) p 587 3674 a(ha) m(v) m(e) p 804 3674 a(already) p 1140 3674 a(sho) m(wn) p 1428 3674 a(that) p 1631 3674 a(the) p 1791 3674 a(sum) p 1990 3674 a(in) p 2096 3674 a(the) p 2256 3674 a(righ) m(t) p 2483 3674 a(hand) p 2719 3674 a(side) p 2907 3674 a(of) p 3010 3674 a(\(5.14\)) p 3284 3674 a(is) p 3374 3674 a(w) m (ell-de\014ned,) 0 3795 y(since) p 237 3795 a(it) p 332 3795 a(equals) p 626 3795 a(to) p Fn 743 3795 a(E) p Fo 821 3795 a(\(\004) p Fn(u) p Fo(\)) p 1048 3795 a(and) p Fn 1236 3795 a(E) p Fo 1344 3795 a(and) p 1532 3795 a(\004) p 1627 3795 a(is) p 1723 3795 a(w) m(ell-de\014ned) p 2255 3795 a(b) m(y) p 2389 3795 a(Lemma) p 2735 3795 a(5.2.) p 2929 3795 a(Put) p Fn 3118 3795 a(u) p Fl 3174 3810 a(0) p Fo 3241 3795 a(=) p Fn 3344 3795 a(u) p Fk 3418 3795 a(\000) p Fn 3513 3795 a(E) p Fo 3591 3795 a(\004) p Fn(u) p Fo(.) p 3781 3795 a(By) 0 3915 y(\(iii\)) p 190 3915 a(of) p 301 3915 a(Lemma) p 649 3915 a(5.2,) p 833 3915 a(w) m(e) p 976 3915 a(ha) m(v) m(e) p Fn 1201 3915 a(u) p Fl 1257 3930 a(0) p Fk 1324 3915 a(2) p Fn 1418 3915 a(D) p Fo 1502 3915 a(\() p 1540 3837 V Fn(L) p Fm 1606 3930 a(N) p Fo 1673 3915 a(\).) p 1781 3915 a(If) p Fn 1879 3915 a(u) p Fl 1935 3930 a(0) p Fo 1996 3915 a(+) p Fn 2093 3915 a(E) p Fo 2171 3915 a(\() p Fq(c) p Fo(\)) p 2329 3915 a(=) p Fn 2438 3915 a(u) p Fg 2494 3879 a(0) p Fl 2494 3940 a(0) p Fo 2554 3915 a(+) p Fn 2652 3915 a(E) p Fo 2730 3915 a(\() p Fq(c) p Fg 2818 3879 a(0) p Fo 2841 3915 a(\),) p Fn 2939 3915 a(u) p Fl 2995 3930 a(0) p Fo 3034 3915 a(,) p Fn 3093 3915 a(u) p Fg 3149 3879 a(0) p Fl 3149 3940 a(0) p Fk 3216 3915 a(2) p Fn 3310 3915 a(D) p Fo 3394 3915 a(\() p 3432 3837 V Fn(L) p Fm 3498 3930 a(N) p Fo 3565 3915 a(\)) p 3636 3915 a(and) p Fq 3825 3915 a(c) p Fo(,) p Fq 0 4035 a(c) p Fg 50 3999 a(0) p Fk 101 4035 a(2) p Fj 195 4035 a(C) p Fl 283 3993 a(4) p Fm(N) p Fo 386 4035 a(,) p 445 4035 a(then) p 668 4035 a(w) m(e) p 811 4035 a(ha) m(v) m(e) p Fq 1149 4217 a(c) p Fk 1221 4217 a(\000) p Fq 1321 4217 a(c) p Fg 1371 4175 a(0) p Fo 1422 4217 a(=) p 1525 4217 a(\004) p Fn(E) p Fo 1668 4217 a(\() p Fq(c) p Fk 1778 4217 a(\000) p Fq 1878 4217 a(c) p Fg 1928 4175 a(0) p Fo 1951 4217 a(\)) p 2017 4217 a(=) p 2120 4217 a(\004\() p Fn(u) p Fg 2279 4175 a(0) p Fl 2279 4241 a(0) p Fk 2340 4217 a(\000) p Fn 2440 4217 a(u) p Fl 2496 4232 a(0) p Fo 2535 4217 a(\)) p 2600 4217 a(=) p 2704 4217 a(0) 0 4398 y(b) m(y) p 140 4398 a(\(iv\)) p 331 4398 a(and) p 525 4398 a(\(v\)) p 689 4398 a(of) p 804 4398 a(Lemma) p 1157 4398 a(5.2.) p 1365 4398 a(Th) m(us) p 1616 4398 a(w) m(e) p 1764 4398 a(ha) m(v) m(e) p Fq 1993 4398 a(c) p Fo 2078 4398 a(=) p Fq 2189 4398 a(c) p Fg 2239 4361 a(0) p Fo 2299 4398 a(and) p Fn 2493 4398 a(u) p Fl 2549 4413 a(0) p Fo 2622 4398 a(=) p Fn 2733 4398 a(u) p Fg 2789 4361 a(0) p Fl 2789 4422 a(0) p Fo 2828 4398 a(,) p 2893 4398 a(so) p 3017 4398 a(the) p 3189 4398 a(decomp) s(osition) p 3836 4398 a(is) 0 4518 y(unique) p 320 4518 a(and) p 512 4518 a(the) p 683 4518 a(map) p Fi 912 4486 a(e) p Fo 903 4518 a(\004) p 1003 4518 a(is) p 1104 4518 a(w) m(ell-de\014ned.) p 1684 4518 a(Since) p Fi 1951 4486 a(e) p Fo 1942 4518 a(\004) p Fn(u) p Fo 2095 4518 a(=) p 2203 4518 a(0) p 2287 4518 a(implies) p Fn 2621 4518 a(u) p Fo 2709 4518 a(=) p Fn 2817 4518 a(u) p Fl 2873 4533 a(0) p Fo 2936 4518 a(+) p Fn 3036 4518 a(E) p Fo 3114 4518 a(\(\004) p Fn(u) p Fo(\)) p 3343 4518 a(=) p 3451 4518 a(0,) p 3563 4518 a(w) m(e) p 3710 4518 a(ha) m(v) m(e) 0 4638 y(that) p Fi 221 4606 a(e) p Fo 211 4638 a(\004) p 309 4638 a(is) p 407 4638 a(injectiv) m(e.) p 832 4638 a(Moreo) m(v) m(er,) p 1288 4638 a(for) p 1437 4638 a(an) m(y) p 1621 4638 a(\() p Fn(u) p Fl 1715 4653 a(0) p Fn 1754 4638 a(;) p Fq 1798 4638 a(c) p Fo(\)) p Fk 1913 4638 a(2) p Fn 2007 4638 a(D) p Fo 2091 4638 a(\() p 2129 4560 V Fn(L) p Fm 2195 4653 a(N) p Fo 2262 4638 a(\)) p Fk 2322 4638 a(\010) p Fj 2422 4638 a(C) p Fl 2510 4596 a(4) p Fm(N) p Fo 2613 4638 a(,) p 2672 4638 a(w) m(e) p 2816 4638 a(ha) m(v) m(e) p Fi 551 4787 a(e) p Fo 542 4819 a(\004\() p Fn(u) p Fl 701 4834 a(0) p Fo 762 4819 a(+) p Fn 860 4819 a(E) p Fo 938 4819 a(\() p Fq(c) p Fo(\)\)) p 1185 4819 a(=) p 1344 4819 a(\() p Fn 1381 4819 a(u) p Fl 1437 4834 a(0) p Fo 1499 4819 a(+) p Fn 1597 4819 a(E) p Fo 1675 4819 a(\() p Fq(c) p Fo(\)) p Fk 1822 4819 a(\000) p Fn 1922 4819 a(E) p Fo 2017 4819 a(\(\004) p 2137 4819 a(\() p Fn 2174 4819 a(u) p Fl 2230 4834 a(0) p Fo 2291 4819 a(+) p Fn 2389 4819 a(E) p Fo 2467 4819 a(\() p Fq(c) p Fo(\)\)\)) p Fn 2686 4819 a(;) p Fo 2730 4819 a(\004) p 2812 4819 a(\() p Fn 2849 4819 a(u) p Fl 2905 4834 a(0) p Fo 2966 4819 a(+) p Fn 3064 4819 a(E) p Fo 3142 4819 a(\() p Fq(c) p Fo(\)\)\)) 1185 4965 y(=) p 1344 4965 a(\() p Fn(u) p Fl 1438 4980 a(0) p Fn 1476 4965 a(;) p Fq 1520 4965 a(c) p Fo(\)) 1902 5214 y(31) p eop %%Page: 32 32 32 31 bop Fo 0 631 a(b) m(y) p 140 631 a(\(iv\)) p 331 631 a(and) p 525 631 a(\(v\)) p 689 631 a(of) p 804 631 a(Lemma) p 1156 631 a(5.2.) p 1364 631 a(Th) m(us) p Fi 1625 599 a(e) p Fo 1615 631 a(\004) p 1717 631 a(is) p 1819 631 a(bijectiv) m(e) p 2211 631 a(and) p 2405 631 a(the) p 2577 631 a(in) m(v) m(erse) p 2904 631 a(map) p 3125 631 a(is) p 3227 631 a(giv) m(en) p 3486 631 a(b) m(y) p 3626 631 a(\(5.16\).) 0 751 y(By) p 153 751 a(\(5.8\)) p 386 751 a(and) p 576 751 a(\(5.10\),) p 885 751 a(w) m(e) p 1028 751 a(ha) m(v) m(e) p Fk 1004 970 a(jj) p Fn(u) p Fl 1116 985 a(0) p Fk 1154 970 a(jj) p Fl 1210 929 a(2) p Fm 1210 994 a(N) p Fo 1299 970 a(+) p Fk 1397 970 a(jj) p Fo(\004) p Fn(u) p Fk(jj) p Fl 1630 929 a(2) p Fa 1630 995 a(C) p Fd 1689 976 a(4) p Fe(N) p Fk 1862 970 a(\024) p 2023 970 a(jj) p Fn(u) p Fk 2156 970 a(\000) p Fn 2255 970 a(E) p Fo 2333 970 a(\(\004) p Fn(u) p Fo(\)) p Fk(jj) p Fl 2586 929 a(2) p Fm 2586 994 a(N) p Fo 2675 970 a(+) p Fk 2773 970 a(jj) p Fo(\004) p Fn(u) p Fk(jj) p Fl 3006 929 a(2) p Fa 3006 995 a(C) p Fd 3065 976 a(4) p Fe(N) p Fk 1862 1115 a(\024) p 2023 1115 a(f) p Fo(2\(1) p 2230 1115 a(+) p Fn 2328 1115 a(C) p Fl 2398 1130 a(2) p Fn 2437 1115 a(C) p Fl 2507 1130 a(3) p Fo 2547 1115 a(\)) p 2607 1115 a(+) p Fn 2705 1115 a(C) p Fl 2775 1130 a(2) p Fk 2814 1115 a(gjj) p Fn(u) p Fk(jj) p Fl 3032 1074 a(2) p Fm 3032 1140 a(N) p Fn 3097 1115 a(;) p Fk 747 1260 a(jj) p Fn(u) p Fk(jj) p Fl 915 1219 a(2) p Fm 915 1285 a(N) p Fo 1008 1260 a(=) p Fk 1112 1260 a(jj) p Fn(u) p Fl 1224 1275 a(0) p Fo 1284 1260 a(+) p Fn 1382 1260 a(E) p Fo 1460 1260 a(\(\004) p Fn(u) p Fo(\)) p Fk(jj) p Fl 1713 1219 a(2) p Fm 1713 1285 a(N) p Fk 1862 1260 a(\024) p Fo 2023 1260 a(2\() p Fk(jj) p Fn(u) p Fl 2222 1275 a(0) p Fk 2260 1260 a(jj) p Fl 2316 1219 a(2) p Fm 2316 1285 a(N) p Fo 2405 1260 a(+) p Fn 2503 1260 a(C) p Fl 2573 1275 a(3) p Fk 2612 1260 a(jj) p Fo(\004) p Fn(u) p Fk(jj) p Fl 2845 1219 a(2) p Fa 2845 1285 a(C) p Fd 2904 1266 a(4) p Fe(N) p Fo 2994 1260 a(\)) p Fn(:) p Fo 0 1479 a(Th) m(us) p 251 1479 a(\(5.17\)) p 536 1479 a(holds) p 794 1479 a(for) p Fn 947 1479 a(C) p Fl 1017 1494 a(4) p Fo 1089 1479 a(=) p 1199 1479 a(max) o(\(2\(1) p 1578 1479 a(+) p Fn 1679 1479 a(C) p Fl 1749 1494 a(2) p Fn 1788 1479 a(C) p Fl 1858 1494 a(3) p Fo 1897 1479 a(\)) p 1959 1479 a(+) p Fn 2060 1479 a(C) p Fl 2130 1494 a(2) p Fn 2169 1479 a(;) p Fo 2213 1479 a(2\(1) p 2373 1479 a(+) p Fn 2473 1479 a(C) p Fl 2543 1494 a(3) p Fo 2582 1479 a(\)\).) p 2739 1479 a(This) p 2965 1479 a(implies) p Fi 3309 1446 a(e) p Fo 3299 1479 a(\004) p 3400 1479 a(is) p 3502 1479 a(a) p 3586 1479 a(Banac) m(h) 0 1599 y(space) p 261 1599 a(isomorphism.) p 869 1531 64 4 v 869 1624 4 93 v 929 1624 V 869 1627 64 4 v 146 1826 a(W) p 238 1826 a(e) p 314 1826 a(shall) p 542 1826 a(giv) m(e) p 743 1826 a(a) p 824 1826 a(c) m(haracterization) p 1537 1826 a(of) p 1648 1826 a(self-adjoin) m(t) p 2150 1826 a(extensions) p 2620 1826 a(of) p Fn 2731 1826 a(L) p Fm 2797 1841 a(N) p Fo 2864 1826 a(.) p Fq 146 2028 a(Prop) s(osition) p 752 2028 a(5.4) p Ff 944 2028 a(L) p 1000 2028 a(et) p Fn 1113 2028 a(N) p Fo 1231 2028 a(=) p 1336 2028 a(1) p Fn(;) p Fo 1429 2028 a(2) p Fn(;) p 1522 2028 a(:) p 1566 2028 a(:) p 1610 2028 a(:) p Ff 1671 2028 a(or) p Fn 1798 2028 a(N) p Fo 1915 2028 a(=) p Fk 2020 2028 a(1) p Ff(.) p 2197 2028 a(Supp) p 2399 2028 a(ose) p 2569 2028 a(that) p 2769 2028 a(Assumption) p 3306 2028 a(1) p 3391 2028 a(holds.) p 3683 2028 a(Then) 0 2148 y(the) p 162 2148 a(fol) p 272 2148 a(lowing) p 575 2148 a(holds:) 146 2268 y(\(i\)) p 291 2268 a(L) p 347 2268 a(et) p Fn 459 2268 a(M) p Ff 599 2268 a(b) p 639 2268 a(e) p 718 2268 a(a) p 803 2268 a(b) p 843 2268 a(ounde) p 1090 2268 a(d) p 1174 2268 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a(J) p Fo 573 3099 a(=) p Fi 677 2854 a(0) 677 3000 y(B) 677 3050 y(B) 677 3099 y(B) 677 3153 y(@) p Fn 791 2918 a(J) p Fm 845 2933 a(\013) p Fd 890 2942 a(1) p Fn 1042 2918 a(O) p 1233 2918 a(:) p 1277 2918 a(:) p 1321 2918 a(:) p 1472 2918 a(O) 821 3038 y(O) p 1012 3038 a(J) p Fm 1066 3053 a(\013) p Fd 1111 3062 a(2) p Fn 1233 3038 a(:) p 1277 3038 a(:) p 1321 3038 a(:) p 1472 3038 a(O) p Ff 804 3159 a(.) p 847 3159 a(.) p 891 3159 a(.) p 935 3159 a(.) p 979 3159 a(.) p 1023 3159 a(.) p 1067 3159 a(.) p 1110 3159 a(.) p 1154 3159 a(.) p 1198 3159 a(.) p 1242 3159 a(.) p 1286 3159 a(.) p 1330 3159 a(.) p 1373 3159 a(.) p 1417 3159 a(.) p 1461 3159 a(.) p 1505 3159 a(.) p 1549 3159 a(.) p Fn 821 3279 a(O) p 1042 3279 a(O) p 1233 3279 a(:) p 1277 3279 a(:) p 1321 3279 a(:) p 1430 3279 a(J) p Fm 1484 3294 a(\013) p Fe 1529 3305 a(N) p Fi 1633 2854 a(1) 1633 3000 y(C) 1633 3050 y(C) 1633 3099 y(C) 1633 3153 y(A) p Fn 1722 3099 a(;) p 1870 3099 a(J) p Fm 1924 3114 a(\013) p Fo 2002 3099 a(=) p 2105 3099 a(4) p Fn(\031) p Fi 2229 2854 a(0) 2229 3000 y(B) 2229 3050 y(B) 2229 3099 y(B) 2229 3153 y(@) p Fo 2436 2918 a(0) p Fn 2660 2918 a(\013) p Fk 2745 2918 a(\000) p Fo 2844 2918 a(1) p 3022 2918 a(0) p 3206 2918 a(0) 2344 3038 y(1) p Fk 2414 3038 a(\000) p Fn 2514 3038 a(\013) p Fo 2752 3038 a(0) p 3022 3038 a(0) p 3206 3038 a(0) 2436 3159 y(0) p 2752 3159 a(0) p 3022 3159 a(0) p Fn 3199 3159 a(\013) p Fo 2436 3279 a(0) p 2752 3279 a(0) p Fk 2976 3279 a(\000) p Fn(\013) p Fo 3206 3279 a(0) p Fi 3303 2854 a(1) 3303 3000 y(C) 3303 3050 y(C) 3303 3099 y(C) 3303 3153 y(A) p Fn 3393 3099 a(:) p Ff 0 3458 a(De\014ne) p 308 3458 a(an) p 447 3458 a(op) p 542 3458 a(er) p 623 3458 a(ator) p Fn 831 3458 a(H) p Fm 920 3422 a(M) 912 3483 y(N) p Ff 1033 3458 a(by) p Fn 1203 3677 a(D) p Fo 1287 3677 a(\() p Fn(H) p Fm 1414 3636 a(M) 1406 3701 y(N) p Fo 1492 3677 a(\)) p 1613 3677 a(=) p Fn 1772 3677 a(D) p Fo 1856 3677 a(\() p 1894 3599 134 4 v Fn(L) p Fm 1960 3692 a(N) p Fo 2027 3677 a(\)) p Fk 2087 3677 a(\010) p Fm 2164 3692 a(al) q(g) p Fo 2286 3677 a(Ran) p Fn 2478 3677 a(E) p 2556 3677 a(M) p 2655 3677 a(;) p Fo 3653 3677 a(\(5.19\)) p Fn 1307 3822 a(H) p Fm 1396 3781 a(M) 1388 3847 y(N) p Fn 1475 3822 a(u) p Fo 1613 3822 a(=) p Fk 1772 3822 a(L) p Fm 1841 3837 a(N) p Fn 1908 3822 a(u;) p 2112 3822 a(u) p Fk 2195 3822 a(2) p Fn 2289 3822 a(D) p Fo 2373 3822 a(\() p Fn(H) p Fm 2500 3781 a(M) 2492 3847 y(N) p Fo 2578 3822 a(\)) p Fn(:) p Ff 0 4040 a(Then,) p Fn 284 4040 a(H) p Fm 373 4004 a(M) 365 4065 y(N) p Ff 486 4040 a(is) p 591 4040 a(a) p 676 4040 a(self-adjoint) p 1181 4040 a(extension) p 1612 4040 a(of) p Fn 1727 4040 a(L) p Fm 1793 4055 a(N) p Ff 1860 4040 a(.) 146 4161 y(\(ii\)) p 325 4161 a(L) p 381 4161 a(et) p Fn 497 4161 a(H) p Fm 578 4176 a(N) p Ff 684 4161 a(b) p 724 4161 a(e) p 808 4161 a(any) p 999 4161 a(self-adjoint) p 1508 4161 a(extension) p 1943 4161 a(of) p Fn 2062 4161 a(L) p Fm 2128 4176 a(N) p Ff 2196 4161 a(.) p 2283 4161 a(Then,) p 2572 4161 a(ther) p 2735 4161 a(e) p 2819 4161 a(exists) p 3089 4161 a(a) p 3178 4161 a(b) p 3218 4161 a(ounde) p 3465 4161 a(d) p 3553 4161 a(op) p 3648 4161 a(er) p 3729 4161 a(ator) p Fn 0 4281 a(M) p Ff 140 4281 a(on) p Fq 279 4281 a(C) p Fl 360 4245 a(4) p Fm(N) p Ff 497 4281 a(satisfying) p 931 4281 a(\(5.18\)) p 1225 4281 a(and) p Fn 1414 4281 a(H) p Fm 1503 4245 a(M) 1495 4306 y(N) p Fo 1609 4281 a(=) p Fn 1713 4281 a(H) p Fm 1794 4296 a(N) p Ff 1896 4281 a(in) p 2015 4281 a(the) p 2177 4281 a(op) p 2272 4281 a(er) p 2353 4281 a(ator) p 2561 4281 a(sense.) 146 4402 y(\(iii\)) p 352 4402 a(F) p 409 4402 a(or) p 536 4402 a(two) p 719 4402 a(b) p 759 4402 a(ounde) p 1006 4402 a(d) p 1092 4402 a(op) p 1187 4402 a(er) p 1268 4402 a(ators) p Fn 1517 4402 a(M) p Ff 1658 4402 a(and) p Fn 1849 4402 a(M) p Fg 1953 4365 a(0) p Ff 2014 4402 a(satisfying) p 2449 4402 a(\(5.18\),) p 2775 4402 a(we) p 2921 4402 a(have) p 3147 4402 a(that) p 3347 4402 a(the) p 3511 4402 a(c) p 3551 4402 a(ondition) p Fn 0 4522 a(H) p Fm 89 4486 a(M) 81 4547 y(N) p Fo 195 4522 a(=) p Fn 299 4522 a(H) p Fm 388 4486 a(M) p Fb 463 4462 a(0) p Fm 380 4547 a(N) p Ff 523 4522 a(is) p 628 4522 a(e) p 668 4522 a(quivalent) p 1081 4522 a(to) p 1198 4522 a(the) p 1360 4522 a(c) p 1400 4522 a(ondition) p Fo 1786 4522 a(Ran) p Fn 1978 4522 a(M) p Fo 2110 4522 a(=) p 2214 4522 a(Ran) p Fn 2405 4522 a(M) p Fg 2509 4486 a(0) p Ff 2533 4522 a(.) 146 4642 y(\(iv\)) p 336 4642 a(A) n(n) p 495 4642 a(element) p Fn 856 4642 a(u) p Ff 947 4642 a(in) p Fn 1066 4642 a(D) p Fo 1150 4642 a(\() p Fn(L) p Fg 1254 4606 a(\003) p Fm 1254 4667 a(N) p Fo 1322 4642 a(\)) p Ff 1394 4642 a(b) p 1434 4642 a(elongs) p 1728 4642 a(to) p Fn 1845 4642 a(D) p Fo 1929 4642 a(\() p Fn(H) p Fm 2056 4606 a(M) 2048 4667 y(N) p Fo 2134 4642 a(\)) p Ff 2207 4642 a(if) p 2302 4642 a(and) p 2491 4642 a(only) p 2703 4642 a(if) p Fo 2797 4642 a(\004) p Fn(u) p Fk 2946 4642 a(2) p Fo 3040 4642 a(Ran) p Fn 3231 4642 a(M) p Ff 3335 4642 a(.) p Fo 0 4844 a(Remark.) p 409 4844 a(Prop) s(osition) p 935 4844 a(5.4) p 1094 4844 a(is) p 1193 4844 a(a) p 1276 4844 a(paraphrase) p 1776 4844 a(of) p 1889 4844 a(the) p 2058 4844 a(prop) s(osition) p 2572 4844 a(in) p 2687 4844 a([Ex-St-Vy) p 3105 4844 a(,) p 3167 4844 a(page) p 3396 4844 a(2159],) p 3680 4844 a(when) p Fn 0 4965 a(N) p Fo 116 4965 a(=) p 220 4965 a(1.) 1902 5214 y(32) p eop %%Page: 33 33 33 32 bop Ff 146 631 a(Pr) p 248 631 a(o) p 293 631 a(of.) p Fo 435 631 a(First) p 673 631 a(w) m(e) p 817 631 a(pro) m(v) m(e) p 1080 631 a(\(iv\).) p 1304 631 a(This) p 1527 631 a(follo) m(ws) p 1847 631 a(from) p 2078 631 a(the) p 2246 631 a(equalit) m(y) p Fi 986 824 a(e) p Fo 976 857 a(\004) p Fi 1058 760 a(\020) p Fn 1107 857 a(D) p Fo 1191 857 a(\() p 1229 779 134 4 v Fn(L) p Fm 1295 872 a(N) p Fo 1363 857 a(\)\)) p Fk 1461 857 a(\010) p Fm 1538 872 a(al) q(g) p Fo 1660 857 a(Ran) p Fn 1851 857 a(E) p 1929 857 a(M) p Fi 2033 760 a(\021) p Fo 2111 857 a(=) p Fn 2215 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a(0) p Fo 2354 1581 a(\)) p Fa 2392 1600 a(C) p Fd 2451 1581 a(4) p Fe(N) p Fo 3653 1581 a(\(5.20\)) 0 1753 y(for) p 152 1753 a(an) m(y) p Fq 339 1753 a(c) p Fn(;) p Fq 433 1753 a(c) p Fg 483 1717 a(0) p Fk 537 1753 a(2) p Fj 636 1753 a(C) p Fl 724 1711 a(4) p Fm(N) p Fo 827 1753 a(,) p 890 1753 a(since) p Fq 1131 1753 a(c) p Fo 1213 1753 a(=) p Fn 1321 1753 a(\016) p Fm 1364 1768 a(k) p Fn 1407 1753 a(;) p Fq 1451 1753 a(c) p Fg 1501 1717 a(0) p Fo 1556 1753 a(=) p Fn 1664 1753 a(\016) p Fm 1707 1768 a(k) p Fb 1746 1749 a(0) p Fo 1807 1753 a(satis\014es) p 2169 1753 a(\(5.20\)) p 2454 1753 a(for) p 2605 1753 a(an) m(y) p Fn 2792 1753 a(k) s(;) p 2890 1753 a(k) p Fg 2944 1717 a(0) p Fo 2999 1753 a(=) p 3107 1753 a(1) p Fn(;) p 3200 1753 a(:) p 3244 1753 a(:) p 3288 1753 a(:) p 3331 1753 a(;) p Fo 3375 1753 a(4) p Fn(N) p Fo 3547 1753 a(b) m(y) p 3685 1753 a(\(i\)) p 3823 1753 a(of) 0 1873 y(Lemma) p 351 1873 a(5.1) p 512 1873 a(\() p Fn(\016) p Fm 593 1888 a(k) p Fo 671 1873 a(is) p 772 1873 a(the) p 943 1873 a(elemen) m(t) p 1307 1873 a(of) p Fj 1421 1873 a(C) p Fl 1509 1831 a(4) p Fm(N) p Fo 1648 1873 a(satisfying) p Fn 2085 1873 a(\016) p Fm 2128 1888 a(k) p Fo 2171 1873 a(\() p Fn(n) p Fo(\)) p 2338 1873 a(=) p Fn 2447 1873 a(\016) p Fm 2490 1888 a(k) r(n) p Fo 2576 1873 a(\).) p 2694 1873 a(F) p 2750 1873 a(or) p 2872 1873 a(an) p 3010 1873 a(elemen) m(t) p Fn 3374 1873 a(u) p Fo 3465 1873 a(in) p Fn 3582 1873 a(D) p Fo 3666 1873 a(\() p Fn(L) p Fg 3770 1837 a(\003) p Fm 3770 1898 a(N) p Fo 3837 1873 a(\),) 0 1994 y(w) m(e) p 144 1994 a(ha) m(v) m(e) p 369 1994 a(an) p 504 1994 a(equiv) p 726 1994 a(alence) p 1019 1994 a(relation) p 1377 1994 a(\(the) p 1582 1994 a(note) p 1799 1994 a(`) p Fk(,) p Fo(') p 1986 1994 a(means) p 2284 1994 a(the) p 2452 1994 a(equiv) p 2674 1994 a(alence\)) p Fn 716 2207 a(u) p Fk 800 2207 a(2) p Fn 894 2207 a(D) p Fo 978 2207 a(\(\() p Fn(H) p Fm 1143 2166 a(M) 1135 2231 y(N) p Fo 1221 2207 a(\)) p Fg 1259 2166 a(\003) p Fo 1298 2207 a(\)) p Fk 1419 2207 a(,) p Fo 1602 2207 a([) p Fn(u;) p 1729 2207 a(v) p Fo 1780 2207 a(]) p Fm 1807 2222 a(N) p Fo 1901 2207 a(=) p 2004 2207 a(0) p 2086 2207 a(for) p 2235 2207 a(ev) m(ery) p Fn 2493 2207 a(v) p Fk 2572 2207 a(2) p Fn 2666 2207 a(D) p Fo 2750 2207 a(\() p Fn(H) p Fm 2877 2166 a(M) 2869 2231 y(N) p Fo 2955 2207 a(\)) p Fk 1419 2352 a(,) p Fo 1602 2352 a([) p Fn(u;) p 1729 2352 a(E) p 1807 2352 a(M) p 1911 2352 a(c) p Fo(]) p Fm 1980 2367 a(N) p Fo 2075 2352 a(=) p 2178 2352 a(0) p 2260 2352 a(for) p 2409 2352 a(ev) m(ery) p Fq 2668 2352 a(c) p Fk 2745 2352 a(2) p Fj 2839 2352 a(C) p Fl 2927 2310 a(4) p Fm(N) p Fk 1419 2497 a(,) p Fo 1602 2497 a(\(\004) p Fn(u;) p 1805 2497 a(J) p 1868 2497 a(M) p 1972 2497 a(c) p Fo(\)) p Fa 2052 2516 a(C) p Fd 2111 2497 a(4) p Fe(N) p Fo 2231 2497 a(=) p 2334 2497 a(0) p 2415 2497 a(for) p 2564 2497 a(ev) m(ery) p Fq 2823 2497 a(c) p Fk 2901 2497 a(2) p Fj 2995 2497 a(C) p Fl 3083 2455 a(4) p Fm(N) p Fk 1419 2643 a(,) p Fo 1602 2643 a(\004) p Fn(u) p Fk 1750 2643 a(2) p Fo 1844 2643 a(\(Ran) p Fn 2073 2643 a(J) p 2136 2643 a(M) p Fo 2240 2643 a(\)) p Fg 2278 2601 a(?) p Fk 1419 2788 a(,) p Fo 1602 2788 a(\004) p Fn(u) p Fk 1750 2788 a(2) p Fo 1844 2788 a(k) m(er) p Fn 1991 2788 a(M) p Fg 2095 2747 a(\003) p Fn 2135 2788 a(J) n(;) p Fo 3653 2788 a(\(5.21\)) 0 3001 y(where) p 282 3001 a(w) m(e) p 426 3001 a(used) p 649 3001 a(the) p 817 3001 a(fact) p 1009 3001 a([) p Fn(u;) p 1136 3001 a(v) p Fo 1187 3001 a(]) p Fm 1214 3016 a(N) p Fo 1309 3001 a(=) p 1412 3001 a(0) p 1494 3001 a(for) p 1643 3001 a(ev) m(ery) p Fn 1901 3001 a(v) p Fk 1979 3001 a(2) p Fn 2074 3001 a(D) p Fo 2158 3001 a(\() p 2196 2923 V Fn(L) p Fm 2262 3016 a(N) p Fo 2329 3001 a(\)) p 2399 3001 a(in) p 2513 3001 a(the) p 2682 3001 a(second) p 2997 3001 a(equiv) p 3219 3001 a(alence,) p 3539 3001 a(\(5.20\)) p 3821 3001 a(in) 0 3121 y(the) p 164 3121 a(third.) p 445 3121 a(By) p 595 3121 a(\(iv\)) p 778 3121 a(of) p 885 3121 a(this) p 1072 3121 a(prop) s(osition) p 1581 3121 a(and) p 1767 3121 a(the) p 1932 3121 a(assumption) p 2444 3121 a(\(5.18\),) p 2750 3121 a(the) p 2914 3121 a(last) p 3096 3121 a(statemen) m(t) p 3545 3121 a(of) p 3653 3121 a(\(5.21\)) 0 3242 y(is) p 100 3242 a(equiv) p 322 3242 a(alen) m(t) p 565 3242 a(to) p 687 3242 a(the) p 857 3242 a(condition) p Fn 1287 3242 a(u) p Fk 1373 3242 a(2) p Fn 1471 3242 a(D) p Fo 1555 3242 a(\() p Fn(H) p Fm 1682 3206 a(M) 1674 3266 y(N) p Fo 1760 3242 a(\).) p 1874 3242 a(Th) m(us) p 2123 3242 a(w) m(e) p 2269 3242 a(ha) m(v) m(e) p Fn 2496 3242 a(D) p Fo 2580 3242 a(\() p Fn(H) p Fm 2707 3206 a(M) 2699 3266 y(N) p Fo 2785 3242 a(\)) p 2854 3242 a(=) p Fn 2961 3242 a(D) p Fo 3045 3242 a(\(\() p Fn(H) p Fm 3210 3206 a(M) 3202 3266 y(N) p Fo 3288 3242 a(\)) p Fg 3326 3206 a(\003) p Fo 3365 3242 a(\).) p 3479 3242 a(Moreo) m(v) m(er,) 0 3362 y(since) p 239 3284 V Fn 239 3362 a(L) p Fm 305 3377 a(N) p Fk 400 3362 a(\032) p Fn 505 3362 a(H) p Fm 594 3326 a(M) 586 3387 y(N) p Fk 701 3362 a(\032) p Fn 806 3362 a(L) p Fg 872 3326 a(\003) p Fm 872 3387 a(N) p Fo 939 3362 a(,) p 999 3362 a(w) m(e) p 1143 3362 a(ha) m(v) m(e) p 1368 3284 V Fn 1368 3362 a(L) p Fm 1434 3377 a(N) p Fk 1529 3362 a(\032) p Fo 1634 3362 a(\() p Fn(H) p Fm 1761 3326 a(M) 1753 3387 y(N) p Fo 1839 3362 a(\)) p Fg 1877 3326 a(\003) p Fk 1944 3362 a(\032) p Fn 2050 3362 a(L) p Fg 2116 3326 a(\003) p Fm 2116 3387 a(N) p Fo 2183 3362 a(.) p 2254 3362 a(Th) m(us) 1283 3575 y(\() p Fn(H) p Fm 1410 3534 a(M) 1402 3600 y(N) p Fo 1489 3575 a(\)) p Fg 1527 3534 a(\003) p Fn 1566 3575 a(u) p Fo 1649 3575 a(=) p Fn 1753 3575 a(L) p Fg 1819 3534 a(\003) p Fm 1819 3600 a(N) p Fn 1886 3575 a(u) p Fo 1970 3575 a(=) p Fk 2073 3575 a(L) p Fm 2142 3590 a(N) p Fn 2209 3575 a(u) p Fo 2292 3575 a(=) p Fn 2396 3575 a(H) p Fm 2485 3534 a(M) 2477 3600 y(N) p Fn 2563 3575 a(u) p Fo 0 3788 a(for) p 149 3788 a(ev) m(ery) p Fn 406 3788 a(u) p Fk 490 3788 a(2) p Fn 584 3788 a(D) p Fo 668 3788 a(\() p Fn(H) p Fm 795 3752 a(M) 787 3813 y(N) p Fo 873 3788 a(\).) p 981 3788 a(Hence) p Fn 1271 3788 a(H) p Fm 1360 3752 a(M) 1352 3813 y(N) p Fo 1471 3788 a(is) p 1569 3788 a(self-adjoin) m(t.) 146 3909 y(\(ii\)) p 316 3909 a(Let) p Fn 498 3909 a(H) p Fm 579 3924 a(N) p Fo 686 3909 a(b) s(e) p 826 3909 a(an) m(y) p 1018 3909 a(self-adjoin) m(t) p 1527 3909 a(extension) p 1966 3909 a(of) p Fn 2084 3909 a(L) p Fm 2150 3924 a(N) p Fo 2218 3909 a(.) p 2310 3909 a(Then) p Fn 2572 3909 a(D) p Fo 2656 3909 a(\() p Fn(H) p Fm 2775 3924 a(N) p Fo 2842 3909 a(\)) p 2919 3909 a(is) p 3025 3909 a(a) p 3113 3909 a(closed) p 3409 3909 a(subspace) p 3823 3909 a(of) p Fn 0 4029 a(D) p Fo 84 4029 a(\() p Fn(L) p Fg 188 3993 a(\003) p Fm 188 4054 a(N) p Fo 255 4029 a(\)) p 326 4029 a(con) m(taining) p Fn 800 4029 a(D) p Fo 884 4029 a(\() p 922 3951 V Fn(L) p Fm 988 4044 a(N) p Fo 1055 4029 a(\).) 146 4150 y(W) p 238 4150 a(e) p 316 4150 a(sho) m(w) p 559 4150 a(that) p Fn 773 4150 a(V) p Fo 882 4150 a(=) p 988 4150 a(\004) p Fn(D) p Fo 1137 4150 a(\() p Fn(H) p Fm 1256 4165 a(N) p Fo 1323 4150 a(\)) p 1395 4150 a(is) p 1495 4150 a(a) p 1578 4150 a(closed) p 1867 4150 a(subspace) p 2277 4150 a(of) p Fj 2390 4150 a(C) p Fl 2478 4108 a(4) p Fm(N) p Fo 2580 4150 a(.) p 2656 4150 a(Supp) s(ose) p 3040 4150 a(that) p Fk 3254 4150 a(f) p Fn(u) p Fm 3360 4165 a(n) p Fk 3406 4150 a(g) p 3486 4150 a(\032) p Fn 3594 4150 a(D) p Fo 3678 4150 a(\() p Fn(H) p Fm 3797 4165 a(N) p Fo 3864 4150 a(\)) 0 4270 y(and) p 196 4270 a(that) p 414 4270 a(\004) p Fn(u) p Fm 535 4285 a(n) p Fk 620 4270 a(!) p Fq 758 4270 a(c) p Fk 846 4270 a(2) p Fn 951 4270 a(C) p Fl 1028 4234 a(4) p Fm(N) p Fo 1131 4270 a(.) p 1221 4270 a(Then) p Fn 1482 4270 a(E) p Fo 1560 4270 a(\(\004) p Fn(u) p Fm 1719 4285 a(n) p Fo 1765 4270 a(\)) p Fk 1842 4270 a(!) p Fn 1980 4270 a(E) p Fo 2058 4270 a(\() p Fq(c) p Fo(\)) p 2223 4270 a(in) p Fn 2343 4270 a(D) p Fo 2427 4270 a(\() p Fn(L) p Fg 2531 4234 a(\003) p Fm 2531 4295 a(N) p Fo 2598 4270 a(\).) p 2726 4270 a(Since) p Fn 2987 4270 a(u) p Fm 3043 4285 a(n) p Fk 3116 4270 a(\000) p Fn 3220 4270 a(E) p Fo 3298 4270 a(\004) p Fn(u) p Fm 3419 4285 a(n) p Fk 3504 4270 a(2) p Fn 3609 4270 a(D) p Fo 3693 4270 a(\() p 3731 4192 V Fn(L) p Fm 3797 4285 a(N) p Fo 3864 4270 a(\)) 0 4390 y(b) m(y) p 145 4390 a(\(iii\)) p 345 4390 a(of) p 466 4390 a(Lemma) p 824 4390 a(5.2,) p 1020 4390 a(w) m(e) p 1174 4390 a(ha) m(v) m(e) p Fn 1408 4390 a(E) p Fo 1486 4390 a(\(\004) p Fn(u) p Fm 1645 4405 a(n) p Fo 1692 4390 a(\)) p 1774 4390 a(=) p Fn 1895 4390 a(u) p Fm 1951 4405 a(n) p Fk 2026 4390 a(\000) p Fo 2132 4390 a(\() p Fn(u) p Fm 2226 4405 a(n) p Fk 2301 4390 a(\000) p Fn 2408 4390 a(E) p Fo 2486 4390 a(\(\004) p Fn(u) p Fm 2645 4405 a(n) p Fo 2691 4390 a(\)\)) p Fk 2811 4390 a(2) p Fn 2922 4390 a(D) p Fo 3006 4390 a(\() p Fn(H) p Fm 3125 4405 a(N) p Fo 3192 4390 a(\).) p 3330 4390 a(Since) p Fn 3594 4390 a(D) p Fo 3678 4390 a(\() p Fn(H) p Fm 3797 4405 a(N) p Fo 3864 4390 a(\)) 0 4511 y(is) p 106 4511 a(a) p 196 4511 a(closed) p 491 4511 a(subspace) p 907 4511 a(in) p Fn 1029 4511 a(D) p Fo 1113 4511 a(\() p Fn(L) p Fg 1217 4475 a(\003) p Fm 1217 4535 a(N) p Fo 1284 4511 a(\),) p 1392 4511 a(w) m(e) p 1544 4511 a(ha) m(v) m(e) p Fn 1777 4511 a(E) p Fo 1855 4511 a(\() p Fq(c) p Fo(\)) p Fk 2022 4511 a(2) p Fn 2130 4511 a(D) p Fo 2214 4511 a(\() p Fn(H) p Fm 2333 4526 a(N) p Fo 2399 4511 a(\).) p 2532 4511 a(By) p 2693 4511 a(\(iv\)) p 2888 4511 a(of) p 3007 4511 a(Lemma) p 3364 4511 a(5.2,) p 3558 4511 a(w) m(e) p 3710 4511 a(ha) m(v) m(e) p Fq 0 4631 a(c) p Fo 77 4631 a(=) p 181 4631 a(\004) p Fn(E) p Fo 324 4631 a(\() p Fq(c) p Fo(\)) p Fk 477 4631 a(2) p Fn 572 4631 a(V) p Fo 650 4631 a(.) p 721 4631 a(Hence) p Fn 1010 4631 a(V) p Fo 1122 4631 a(is) p 1220 4631 a(closed.) 146 4752 y(Th) m(us) p 393 4752 a(the) p 561 4752 a(follo) m(wing) p 973 4752 a(decomp) s(osition) p 1616 4752 a(holds:) p Fi 1456 4932 a(e) p Fo 1447 4965 a(\004) p Fn(D) p Fo 1596 4965 a(\() p Fn(H) p Fm 1715 4980 a(N) p Fo 1782 4965 a(\)) p 1847 4965 a(=) p Fn 1951 4965 a(D) p Fo 2035 4965 a(\() p 2073 4887 V Fn(L) p Fm 2139 4980 a(N) p Fo 2206 4965 a(\)) p Fk 2266 4965 a(\010) p Fn 2366 4965 a(V) p 2428 4965 a(:) p Fo 1902 5214 a(33) p eop %%Page: 34 34 34 33 bop Fo 0 631 a(Let) p Fn 175 631 a(M) p Fo 312 631 a(b) s(e) p 445 631 a(the) p 613 631 a(orthogonal) p 1101 631 a(pro) p 1247 631 a(jection) p 1564 631 a(from) p Fj 1794 631 a(C) p Fl 1883 589 a(4) p Fm(N) p Fo 2018 631 a(on) m(to) p Fn 2237 631 a(V) p Fo 2316 631 a(.) p 2386 631 a(Then) p 2641 631 a(Ran) p Fn 2832 631 a(M) p Fo 2965 631 a(=) p Fn 3068 631 a(V) p Fo 3179 631 a(is) p 3277 631 a(closed) p 3565 631 a(and) p Fn 1118 847 a(D) p Fo 1202 847 a(\() p Fn(H) p Fm 1329 806 a(M) 1321 872 y(N) p Fo 1407 847 a(\)) p 1473 847 a(=) p Fi 1586 815 a(e) p Fo 1576 847 a(\004) p Fg 1641 806 a(\000) p Fl(1) p Fi 1752 751 a(\020) p Fn 1802 847 a(D) p Fo 1886 847 a(\() p 1924 769 134 4 v Fn(L) p Fm 1990 862 a(N) p Fo 2057 847 a(\)) p Fk 2117 847 a(\010) p Fn 2217 847 a(V) p Fi 2295 751 a(\021) p Fo 2373 847 a(=) p Fn 2476 847 a(D) p Fo 2560 847 a(\() p Fn(H) p Fm 2679 862 a(N) p Fo 2746 847 a(\)) 0 1083 y(b) m(y) p 136 1083 a(de\014nition) p 569 1083 a(\(5.19\)) p 851 1083 a(and) p 1041 1083 a(the) p 1209 1083 a(form) p 1439 1083 a(of) p 1551 1083 a(the) p 1719 1083 a(in) m(v) m(erse) p 2042 1083 a(map) p 2259 1083 a(\(5.16\).) p 2579 1083 a(Th) m(us) p Fn 2826 1083 a(H) p Fm 2907 1098 a(N) p Fo 3002 1083 a(=) p Fn 3105 1083 a(H) p Fm 3194 1046 a(M) 3186 1107 y(N) p Fo 3273 1083 a(.) p 3344 1083 a(Moreo) m(v) m(er,) p 3799 1083 a(b) m(y) 0 1203 y(\(iv\),) p 214 1203 a(\(5.21\)) p 496 1203 a(and) p 686 1203 a(the) p 854 1203 a(self-adjoin) m(tness) p 1530 1203 a(of) p Fn 1641 1203 a(H) p Fm 1722 1218 a(N) p Fo 1789 1203 a(,) p 1849 1203 a(w) m(e) p 1992 1203 a(ha) m(v) m(e) 958 1419 y(Ran) p Fn 1149 1419 a(M) p Fo 1281 1419 a(=) p 1385 1419 a(\004) p Fn(D) p Fo 1534 1419 a(\() p Fn(H) p Fm 1661 1378 a(M) 1653 1444 y(N) p Fo 1739 1419 a(\)) p 1805 1419 a(=) p 1908 1419 a(\004) p Fn(D) p Fo 2057 1419 a(\(\() p Fn(H) p Fm 2222 1378 a(M) 2214 1444 y(N) p Fo 2300 1419 a(\)) p Fg 2338 1378 a(\003) p Fo 2378 1419 a(\)) p 2443 1419 a(=) p 2547 1419 a(Ker) p Fn 2721 1419 a(M) p Fg 2825 1378 a(\003) p Fn 2865 1419 a(J) n(:) p Fo 0 1635 a(Th) m(us) p 247 1635 a(\(5.18\)) p 529 1635 a(holds.) 146 1756 y(\(iii\)) p 336 1756 a(By) p 489 1756 a(de\014nition,) 1012 1972 y(Ran) p Fn 1203 1972 a(M) p Fo 1336 1972 a(=) p 1439 1972 a(Ran) p Fn 1631 1972 a(M) p Fg 1735 1931 a(0) p Fk 829 2117 a(,) p Fn 1012 2117 a(D) p Fo 1096 2117 a(\() p 1134 2039 V Fn(L) p Fm 1200 2132 a(N) p Fo 1267 2117 a(\)) p Fk 1327 2117 a(\010) p Fo 1427 2117 a(Ran) p Fn 1618 2117 a(M) p Fo 1751 2117 a(=) p Fn 1854 2117 a(D) p Fo 1938 2117 a(\() p 1976 2039 V Fn(L) p Fm 2042 2132 a(N) p Fo 2110 2117 a(\)) p Fk 2170 2117 a(\010) p Fo 2269 2117 a(Ran) p Fn 2461 2117 a(M) p Fg 2565 2076 a(0) p Fk 829 2275 a(,) p Fi 1022 2242 a(e) p Fo 1012 2275 a(\004) p Fg 1077 2234 a(\000) p Fl(1) p Fi 1188 2178 a(\020) p Fn 1238 2275 a(D) p Fo 1322 2275 a(\() p 1360 2197 V Fn(L) p Fm 1426 2290 a(N) p Fo 1493 2275 a(\)) p Fk 1553 2275 a(\010) p Fo 1653 2275 a(Ran) p Fn 1844 2275 a(M) p Fi 1948 2178 a(\021) p Fo 2026 2275 a(=) p Fi 2139 2242 a(e) p Fo 2129 2275 a(\004) p Fg 2194 2234 a(\000) p Fl(1) p Fi 2305 2178 a(\020) p Fn 2355 2275 a(D) p Fo 2439 2275 a(\() p 2477 2197 V Fn(L) p Fm 2543 2290 a(N) p Fo 2610 2275 a(\)) p Fk 2670 2275 a(\010) p Fo 2770 2275 a(Ran) p Fn 2961 2275 a(M) p Fg 3065 2234 a(0) p Fi 3089 2178 a(\021) p Fk 829 2450 a(,) p Fn 1012 2450 a(D) p Fo 1096 2450 a(\() p Fn(H) p Fm 1223 2409 a(M) 1215 2475 y(N) p Fo 1301 2450 a(\)) p 1367 2450 a(=) p Fn 1470 2450 a(D) p Fo 1554 2450 a(\() p Fn(H) p Fm 1681 2409 a(M) p Fb 1756 2386 a(0) p Fm 1673 2475 a(N) p Fo 1782 2450 a(\)) p Fn(;) p Fo 0 2666 a(where) p 282 2666 a(w) m(e) p 425 2666 a(used) p Fi 658 2634 a(e) p Fo 648 2666 a(\004) p 746 2666 a(is) p 844 2666 a(an) p 979 2666 a(isomorphism.) p 1587 2598 64 4 v 1587 2691 4 93 v 1647 2691 V 1587 2694 64 4 v 0 2866 a(By) p 153 2866 a(the) p 321 2866 a(pro) s(of) p 576 2866 a(of) p 687 2866 a(the) p 855 2866 a(previous) p 1243 2866 a(prop) s(osition,) p 1783 2866 a(w) m(e) p 1926 2866 a(ha) m(v) m(e) p 2151 2866 a(the) p 2319 2866 a(follo) m(wing) p 2731 2866 a(corollary) p 3096 2866 a(.) p Fq 146 3066 a(Corollary) p 644 3066 a(5.5) p Ff 836 3066 a(Supp) p 1038 3066 a(ose) p 1209 3066 a(that) p 1411 3066 a(Assumption) p 1949 3066 a(1) p 2035 3066 a(holds.) p 2331 3066 a(Then,) p 2617 3066 a(the) p 2781 3066 a(map) p Fn 2997 3066 a(M) p Fk 3134 3066 a(7!) p Fn 3265 3066 a(H) p Fm 3354 3030 a(M) 3346 3091 y(N) p Ff 3469 3066 a(is) p 3576 3066 a(an) p 3718 3066 a(one-) 0 3187 y(to-one,) p 331 3187 a(onto) p 553 3187 a(c) p 593 3187 a(orr) p 720 3187 a(esp) p 850 3187 a(ondenc) p 1145 3187 a(e) p 1223 3187 a(fr) p 1289 3187 a(om) p 1453 3187 a(the) p 1615 3187 a(set) p 1767 3187 a(of) p 1882 3187 a(al) p 1962 3187 a(l) p 2021 3187 a(ortho) p 2239 3187 a(gonal) p 2498 3187 a(pr) p 2584 3187 a(oje) p 2704 3187 a(ctions) p Fn 2990 3187 a(M) p Ff 3130 3187 a(on) p Fj 3269 3187 a(C) p Fl 3358 3145 a(4) p Fm(N) p Ff 3495 3187 a(satisfying) p Fo 1566 3403 a(Ker) p Fn 1740 3403 a(M) p 1844 3403 a(J) p Fo 1936 3403 a(=) p 2039 3403 a(Ran) p Fn 2231 3403 a(M) p Fo 3653 3403 a(\(5.22\)) p Ff 0 3619 a(to) p 117 3619 a(the) p 279 3619 a(set) p 431 3619 a(of) p 545 3619 a(al) p 625 3619 a(l) p 685 3619 a(self-adjoint) p 1191 3619 a(extensions) p Fn 1661 3619 a(H) p Fm 1742 3634 a(N) p Ff 1844 3619 a(of) p Fn 1959 3619 a(L) p Fm 2025 3634 a(N) p Ff 2092 3619 a(.) 146 3843 y(Pr) p 248 3843 a(o) p 293 3843 a(of.) p Fo 434 3843 a(When) p Fn 717 3843 a(M) p Fo 853 3843 a(is) p 949 3843 a(an) p 1083 3843 a(orthogonal) p 1570 3843 a(pro) p 1716 3843 a(jection,) p 2059 3843 a(Ran) p Fn 2250 3843 a(M) p Fo 2386 3843 a(is) p 2482 3843 a(closed) p 2769 3843 a(and) p Fn 2957 3843 a(M) p Fg 3061 3807 a(\003) p Fo 3129 3843 a(=) p Fn 3232 3843 a(M) p Fo 3336 3843 a(.) p 3407 3843 a(Th) m(us) p 3653 3843 a(\(5.22\)) 0 3963 y(is) p 104 3963 a(equiv) p 326 3963 a(alen) m(t) p 574 3963 a(to) p 699 3963 a(\(5.18\).) p 1037 3963 a(Hence) p 1333 3963 a(the) p 1507 3963 a(map) p 1730 3963 a(is) p 1834 3963 a(w) m(ell-de\014ned.) p 2424 3963 a(F) p 2480 3963 a(or) p 2605 3963 a(t) m(w) m(o) p 2795 3963 a(orthogonal) p 3289 3963 a(pro) p 3435 3963 a(jections) p Fn 3797 3963 a(M) p Fo 0 4084 a(and) p Fn 192 4084 a(M) p Fg 296 4048 a(0) p Fo 320 4084 a(,) p 383 4084 a(the) p 554 4084 a(condition) p Fn 985 4084 a(M) p Fo 1122 4084 a(=) p Fn 1230 4084 a(M) p Fg 1334 4048 a(0) p Fo 1393 4084 a(is) p 1493 4084 a(equiv) p 1715 4084 a(alen) m(t) p 1959 4084 a(to) p 2081 4084 a(the) p 2252 4084 a(condition) p 2683 4084 a(Ran) p Fn 2874 4084 a(M) p Fo 3011 4084 a(=) p 3119 4084 a(Ran) p Fn 3310 4084 a(M) p Fg 3414 4048 a(0) p Fo 3438 4084 a(.) p 3517 4084 a(Th) m(us) p 3767 4084 a(the) 0 4204 y(injectivit) m(y) p 459 4204 a(of) p 568 4204 a(the) p 735 4204 a(map) p 950 4204 a(follo) m(ws) p 1268 4204 a(from) p 1497 4204 a(\(iii\)) p 1685 4204 a(of) p 1794 4204 a(Prop) s(osition) p 2318 4204 a(5.4.) p 2512 4204 a(The) p 2711 4204 a(surjectivit) m(y) p 3219 4204 a(follo) m(ws) p 3538 4204 a(from) p 3767 4204 a(the) 0 4325 y(pro) s(of) p 255 4325 a(of) p 366 4325 a(\(ii\)) p 528 4325 a(of) p 639 4325 a(Prop) s(osition) p 1164 4325 a(5.4.) p 1359 4257 V 1359 4349 4 93 v 1420 4349 V 1359 4352 64 4 v 0 4548 a(When) p 284 4548 a(the) p 452 4548 a(op) s(erator) p Fn 845 4548 a(M) p Fo 983 4548 a(has) p 1157 4548 a(a) p 1238 4548 a(simple) p 1542 4548 a(form,) p 1799 4548 a(the) p 1967 4548 a(condition) p 2396 4548 a(\(5.18\)) p 2677 4548 a(is) p 2775 4548 a(simpli\014ed) p 3215 4548 a(as) p 3335 4548 a(follo) m(ws:) p Fq 146 4749 a(Lemma) p 543 4749 a(5.6) p Ff 735 4749 a(\(i\)) p 880 4749 a(L) p 936 4749 a(et) p Fn 1048 4749 a(N) p 1164 4749 a(<) p Fk 1268 4749 a(1) p Ff(.) p 1442 4749 a(Then,) p 1726 4749 a(the) p 1888 4749 a(c) p 1928 4749 a(ondition) p 2314 4749 a(\(5.18\)) p 2608 4749 a(is) p 2712 4749 a(e) p 2752 4749 a(quivalent) p 3166 4749 a(to) p Fo 1358 4965 a(rank) p Fn(M) p Fo 1683 4965 a(=) p 1786 4965 a(2) p Fn(N) p 1918 4965 a(;) p 1997 4965 a(M) p Fg 2101 4924 a(\003) p Fn 2141 4965 a(J) p 2204 4965 a(M) p Fo 2336 4965 a(=) p Fn 2440 4965 a(O) s(;) p Fo 3653 4965 a(\(5.23\)) 1902 5214 y(34) p eop %%Page: 35 35 35 34 bop Ff 0 631 a(wher) p 196 631 a(e) p Fo 275 631 a(rank) p Fn 468 631 a(M) p Fo 600 631 a(=) p 704 631 a(dim) p 883 631 a(Ran) p Fn 1074 631 a(M) p Ff 1178 631 a(.) 146 751 y(\(ii\)) p 328 751 a(L) p 384 751 a(et) p Fn 503 751 a(N) p Fo 632 751 a(=) p 748 751 a(1) p Fn(;) p Fo 841 751 a(2) p Fn(;) p 934 751 a(:) p 978 751 a(:) p 1022 751 a(:) p Ff 1089 751 a(or) p Fn 1222 751 a(N) p Fo 1351 751 a(=) p Fk 1467 751 a(1) p Ff(.) p 1663 751 a(Supp) p 1865 751 a(ose) p 2041 751 a(that) p 2247 751 a(Assumption) p 2789 751 a(1) p 2881 751 a(holds.) p 3191 751 a(Supp) p 3393 751 a(ose) p 3569 751 a(that) p 3775 751 a(the) 0 872 y(op) p 95 872 a(er) p 176 872 a(ator) p Fn 383 872 a(M) p Ff 522 872 a(is) p 626 872 a(the) p 788 872 a(\(\014nite) p 1078 872 a(or) p 1204 872 a(in\014nite\)) p 1579 872 a(dir) p 1695 872 a(e) p 1735 872 a(ct) p 1846 872 a(sum) p 2053 872 a(of) p Fo 2167 872 a(4) p Fk 2236 872 a(\002) p Fo 2335 872 a(4) p Ff 2418 872 a(ortho) p 2636 872 a(gonal) p 2894 872 a(pr) p 2980 872 a(oje) p 3100 872 a(ction) p 3346 872 a(matric) p 3619 872 a(es,) p 3767 872 a(i.e.) p Fn 525 1269 a(M) p Fo 658 1269 a(=) p Fi 761 1023 a(0) 761 1169 y(B) 761 1219 y(B) 761 1269 y(B) 761 1322 y(@) p Fn 875 1088 a(M) p Fl 969 1103 a(1) p Fn 1120 1088 a(O) p 1309 1088 a(:) p 1353 1088 a(:) p 1397 1088 a(:) p 1549 1088 a(O) 904 1208 y(O) p 1092 1208 a(M) p Fl 1186 1223 a(2) p Fn 1309 1208 a(:) p 1353 1208 a(:) p 1397 1208 a(:) p 1549 1208 a(O) p Ff 884 1328 a(.) p 928 1328 a(.) p 972 1328 a(.) p 1016 1328 a(.) p 1060 1328 a(.) p 1103 1328 a(.) p 1147 1328 a(.) p 1191 1328 a(.) p 1235 1328 a(.) p 1279 1328 a(.) p 1323 1328 a(.) p 1366 1328 a(.) p 1410 1328 a(.) p 1454 1328 a(.) p 1498 1328 a(.) p 1542 1328 a(.) p 1586 1328 a(.) p 1629 1328 a(.) p Fn 904 1449 a(O) p 1120 1449 a(O) p 1309 1449 a(:) p 1353 1449 a(:) p 1397 1449 a(:) p 1507 1449 a(M) p Fm 1601 1464 a(N) p Fi 1710 1023 a(1) 1710 1169 y(C) 1710 1219 y(C) 1710 1269 y(C) 1710 1322 y(A) p Fn 1799 1269 a(;) p 1947 1269 a(M) p Fm 2041 1284 a(j) p Fo 2106 1269 a(:) p 2161 1269 a(4) p Fk 2231 1269 a(\002) p Fo 2331 1269 a(4) p Fn(;) p 2458 1269 a(M) p Fl 2562 1228 a(2) p Fm 2552 1294 a(j) p Fo 2630 1269 a(=) p Fn 2734 1269 a(M) p Fm 2828 1284 a(j) p Fn 2865 1269 a(;) p 2943 1269 a(M) p Fg 3047 1228 a(\003) p Fm 3037 1294 a(j) p Fo 3115 1269 a(=) p Fn 3219 1269 a(M) p Fm 3313 1284 a(j) p Fn 3350 1269 a(:) p Ff 0 1666 a(Then,) p 284 1666 a(the) p 446 1666 a(c) p 486 1666 a(ondition) p 872 1666 a(\(5.18\)) p 1166 1666 a(is) p 1270 1666 a(e) p 1310 1666 a(quivalent) p 1724 1666 a(to) p Fo 836 1885 a(rank) p Fn 1029 1885 a(M) p Fm 1123 1900 a(j) p Fo 1187 1885 a(=) p 1291 1885 a(2) p Fn(;) p 1418 1885 a(M) p Fm 1512 1900 a(j) p Fn 1549 1885 a(J) p Fm 1603 1900 a(\013) p Fe 1648 1910 a(j) p Fn 1685 1885 a(M) p Fm 1779 1900 a(j) p Fo 1844 1885 a(=) p Fn 1947 1885 a(O) s(;) p Fo 2103 1885 a(for) p 2252 1885 a(ev) m(ery) p Fn 2511 1885 a(j) p Fo 2585 1885 a(=) p 2688 1885 a(1) p Fn(;) p 2781 1885 a(:) p 2825 1885 a(:) p 2869 1885 a(:) p 2912 1885 a(;) p 2956 1885 a(N) p 3039 1885 a(:) p Fo 3653 1885 a(\(5.24\)) p Ff 146 2112 a(Pr) p 248 2112 a(o) p 293 2112 a(of.) p Fo 441 2112 a(\(i\)) p 581 2112 a(When) p Fn 871 2112 a(M) p Fo 1014 2112 a(is) p 1117 2112 a(\014nite,) p 1400 2112 a(Ran) p Fn 1591 2112 a(M) p Fo 1734 2112 a(is) p 1837 2112 a(a) p 1924 2112 a(closed) p 2217 2112 a(subspace.) p 2678 2112 a(The) p 2884 2112 a(equalit) m(y) p Fn 3260 2112 a(M) p Fg 3364 2076 a(\003) p Fn 3404 2112 a(J) p 3467 2112 a(M) p Fo 3609 2112 a(=) p Fn 3721 2112 a(O) p Fo 3836 2112 a(is) 0 2232 y(equiv) p 222 2232 a(alen) m(t) p 463 2232 a(to) 1538 2353 y(Ran) p Fn 1729 2353 a(M) p Fk 1862 2353 a(\032) p Fo 1967 2353 a(Ker) p Fn 2141 2353 a(M) p Fg 2245 2312 a(\003) p Fn 2285 2353 a(J) n(:) p Fo 3653 2353 a(\(5.25\)) 0 2527 y(Since) p Fn 255 2527 a(J) p Fo 350 2527 a(is) p 448 2527 a(a) p 530 2527 a(regular) p 860 2527 a(matrix,) 1526 2647 y(rank) p Fn(M) p Fg 1822 2606 a(\003) p Fn 1863 2647 a(J) p Fo 1954 2647 a(=) p 2057 2647 a(rank) p Fn(M) p 2348 2647 a(:) p Fo 0 2821 a(Th) m(us) 1316 2941 y(dim) p 1495 2941 a(Ker) p Fn 1669 2941 a(M) p Fg 1773 2900 a(\003) p Fn 1813 2941 a(J) p Fo 1904 2941 a(=) p 2008 2941 a(4) p Fn(N) p Fk 2167 2941 a(\000) p Fo 2267 2941 a(rank) p Fn(M) p 2558 2941 a(:) p Fo 0 3115 a(Hence) p 283 3115 a(w) m(e) p 420 3115 a(ha) m(v) m(e) p 638 3115 a(that) p 842 3115 a(the) p 1003 3115 a(condition) p 1425 3115 a(Ran) p Fn 1616 3115 a(M) p Fo 1749 3115 a(=) p 1852 3115 a(Ker) p Fn 2026 3115 a(M) p Fg 2130 3079 a(\003) p Fn 2170 3115 a(J) p Fo 2259 3115 a(is) p 2350 3115 a(equiv) p 2572 3115 a(alen) m(t) p 2807 3115 a(to) p 2919 3115 a(the) p 3080 3115 a(condition) p 3501 3115 a(rank) p Fn 3694 3115 a(M) p Fo 3826 3115 a(=) 0 3236 y(2) p Fn(N) p Fo 137 3236 a(,) p 197 3236 a(under) p 473 3236 a(the) p 641 3236 a(condition) p 1069 3236 a(\(5.25\).) 146 3356 y(\(ii\)) p 309 3356 a(Consider) p 716 3356 a(the) p 884 3356 a(follo) m(wing) p 1296 3356 a(decomp) s(osition) p Fj 682 3616 a(C) p Fl 770 3574 a(4) p Fm(N) p Fk 900 3616 a(3) p Fo 994 3616 a(\() p Fn(c) p Fl 1074 3631 a(1) p Fn 1114 3616 a(;) p 1158 3616 a(:) p 1202 3616 a(:) p 1246 3616 a(:) p 1288 3616 a(;) p 1332 3616 a(c) p Fl 1374 3631 a(4) p Fm(N) p Fo 1477 3616 a(\)) p Fk 1542 3616 a(7!) p Fo 1669 3616 a(\(\() p Fn(c) p Fl 1787 3631 a(4) p Fm(j) p Fg 1855 3631 a(\000) p Fl(3) p Fn 1949 3616 a(;) p 1993 3616 a(c) p Fl 2035 3631 a(4) p Fm(j) p Fg 2103 3631 a(\000) p Fl(2) p Fn 2197 3616 a(;) p 2241 3616 a(c) p Fl 2283 3631 a(4) p Fm(j) p Fg 2351 3631 a(\000) p Fl(1) p Fn 2445 3616 a(;) p 2489 3616 a(c) p Fl 2531 3631 a(4) p Fm(j) p Fo 2602 3616 a(\)\)) p Fm 2678 3568 a(N) 2678 3648 y(j) p Fl 2711 3648 a(=1) p Fk 2832 3616 a(2) p Fm 2956 3541 a(N) p Fk 2949 3616 a(\010) p Fm 2926 3682 a(j) p Fl 2959 3682 a(=1) p Fj 3066 3616 a(C) p Fl 3154 3574 a(4) p Fn 3193 3616 a(:) p Fo 0 3885 a(Since) p Fn 262 3885 a(M) p Fm 356 3900 a(j) p Fo 432 3885 a(\() p Fn 510 3885 a(j) p Fo 596 3885 a(=) p 712 3885 a(1) p Fn(;) p Fo 805 3885 a(2) p Fn(;) p 898 3885 a(:) p 942 3885 a(:) p 986 3885 a(:) p 1028 3885 a(;) p 1072 3885 a(N) p Fo 1160 3885 a(\)) p 1238 3885 a(and) p Fn 1434 3885 a(M) p Fo 1579 3885 a(are) p 1749 3885 a(orthogonal) p 2244 3885 a(pro) p 2390 3885 a(jections,) p 2781 3885 a(w) m(e) p 2932 3885 a(ha) m(v) m(e) p 3164 3885 a(that) p 3383 3885 a(Ran) p Fn 3574 3885 a(M) p Fm 3668 3900 a(j) p Fo 3745 3885 a(and) 0 4005 y(Ran) p Fn 191 4005 a(M) p Fo 329 4005 a(are) p 491 4005 a(closed) p 779 4005 a(and) p 969 4005 a(that) p 1180 4005 a(the) p 1348 4005 a(follo) m(wing) p 1760 4005 a(decomp) s(osition) p 2403 4005 a(holds:) 878 4260 y(Ran) p Fn 1070 4260 a(M) p Fo 1202 4260 a(=) p Fm 1335 4185 a(N) p Fk 1328 4260 a(\010) p Fm 1306 4326 a(j) p Fl 1339 4326 a(=1) p Fo 1445 4260 a(Ran) p Fn 1636 4260 a(M) p Fm 1730 4275 a(j) p Fn 1767 4260 a(;) p Fo 1860 4260 a(k) m(er) p Fn 2007 4260 a(M) p Fg 2111 4218 a(\003) p Fn 2151 4260 a(J) p Fo 2242 4260 a(=) p Fm 2375 4185 a(N) p Fk 2368 4260 a(\010) p Fm 2345 4326 a(j) p Fl 2378 4326 a(=1) p Fo 2484 4260 a(k) m(er) q(\() p Fn(M) p Fm 2746 4275 a(j) p Fo 2783 4260 a(\)) p Fg 2821 4218 a(\003) p Fn 2861 4260 a(J) p Fm 2915 4275 a(\013) p Fe 2960 4285 a(j) p Fn 2997 4260 a(:) p Fo 0 4523 a(Th) m(us) p 247 4523 a(the) p 415 4523 a(condition) p 843 4523 a(Ran) p Fn 1035 4523 a(M) p Fo 1167 4523 a(=) p 1271 4523 a(Ker) p Fn 1444 4523 a(M) p Fg 1548 4487 a(\003) p Fn 1589 4523 a(J) p Fo 1684 4523 a(is) p 1782 4523 a(equiv) p 2004 4523 a(alen) m(t) p 2246 4523 a(to) 957 4742 y(Ran) p Fn 1149 4742 a(M) p Fm 1243 4757 a(j) p Fo 1307 4742 a(=) p 1411 4742 a(Ker\() p Fn(M) p Fm 1700 4757 a(j) p Fo 1737 4742 a(\)) p Fg 1775 4700 a(\003) p Fn 1814 4742 a(J) p Fm 1868 4757 a(\013) p Fe 1913 4767 a(j) p Fo 1982 4742 a(for) p 2131 4742 a(ev) m(ery) p Fn 2390 4742 a(j) p Fo 2464 4742 a(=) p 2567 4742 a(1) p Fn(;) p 2660 4742 a(:) p 2704 4742 a(:) p 2748 4742 a(:) p 2791 4742 a(;) p 2835 4742 a(N) p 2918 4742 a(:) p Fo 0 4960 a(This) p 223 4960 a(is) p 321 4960 a(equiv) p 543 4960 a(alen) m(t) p 784 4960 a(to) p 903 4960 a(\(5.23\),) p 1212 4960 a(b) m(y) p 1348 4960 a(\(i\)) p 1483 4960 a(and) p Fn 1673 4960 a(M) p Fg 1777 4924 a(\003) p Fm 1767 4985 a(j) p Fo 1845 4960 a(=) p Fn 1948 4960 a(M) p Fm 2042 4975 a(j) p Fo 2079 4960 a(.) p 2150 4893 64 4 v 2150 4985 4 93 v 2210 4985 V 2150 4988 64 4 v 1902 5214 a(35) p eop %%Page: 36 36 36 35 bop Fo 146 631 a(W) p 238 631 a(e) p 314 631 a(shall) p 542 631 a(examine) p 925 631 a(some) p 1169 631 a(prop) s(erties) p 1627 631 a(of) p 1739 631 a(the) p 1907 631 a(standard) p 2314 631 a(Aharono) m(v-Bohm) p 3046 631 a(Hamiltonian) p Fn 3607 631 a(H) p Fm 3696 595 a(AB) 3688 656 y(N) p Fo 3809 631 a(.) p Fq 146 834 a(Prop) s(osition) p 752 834 a(5.7) p Ff 944 834 a(L) p 1000 834 a(et) p Fo 1116 834 a(1) p Fk 1198 834 a(\024) p Fn 1310 834 a(N) p 1433 834 a(<) p Fk 1542 834 a(1) p Ff 1680 834 a(or) p Fn 1810 834 a(N) p Fo 1932 834 a(=) p Fk 2042 834 a(1) p Ff(.) p 2227 834 a(L) p 2283 834 a(et) p Fn 2399 834 a(H) p Fm 2488 798 a(AB) 2480 859 y(N) p Ff 2639 834 a(b) p 2679 834 a(e) p 2762 834 a(the) p 2927 834 a(F) p 2984 834 a(rie) p 3095 834 a(drichs) p 3388 834 a(extension) p 3822 834 a(of) p Fn 0 955 a(L) p Fm 66 970 a(N) p Ff 134 955 a(.) p 208 955 a(Supp) p 410 955 a(ose) p 579 955 a(that) p 779 955 a(Assumption) p 1314 955 a(1) p 1399 955 a(holds.) p 1688 955 a(Then,) p 1972 955 a(the) p 2134 955 a(fol) p 2244 955 a(lowing) p 2547 955 a(holds.) 146 1075 y(\(i\)) p 291 1075 a(The) p 490 1075 a(op) p 585 1075 a(er) p 666 1075 a(ator) p 874 1075 a(domain) p Fn 1222 1075 a(D) p Fo 1306 1075 a(\() p Fn(H) p Fm 1433 1039 a(AB) 1425 1100 y(N) p Fo 1546 1075 a(\)) p Ff 1619 1075 a(is) p 1723 1075 a(de) p 1813 1075 a(c) p 1853 1075 a(omp) p 2028 1075 a(ose) p 2158 1075 a(d) p 2241 1075 a(into) p 2443 1075 a(the) p 2605 1075 a(dir) p 2721 1075 a(e) p 2761 1075 a(ct) p 2873 1075 a(sum) p Fn 1054 1336 a(D) p Fo 1138 1336 a(\() p Fn(H) p Fm 1265 1295 a(AB) 1257 1361 y(N) p Fo 1378 1336 a(\)) p 1443 1336 a(=) p Fn 1547 1336 a(D) p Fo 1631 1336 a(\() p 1669 1258 134 4 v Fn(L) p Fm 1735 1351 a(N) p Fo 1802 1336 a(\)) p Fk 1862 1336 a(\010) p Fm 1939 1351 a(al) q(g) 2091 1262 y(N) p Fk 2084 1336 a(\010) p Fm 2061 1402 a(j) p Fl 2094 1402 a(=1) p Fo 2201 1336 a(L) p Fn(:) p Fo(h) p Fn(:) p Fk(f) p Fn(\036) p Fl 2478 1285 a(\() p Fm(j) p Fl 2538 1285 a(\)) 2478 1358 y(0) p Fn 2569 1336 a(;) p 2613 1336 a( ) p Fl 2680 1285 a(\() p Fm(j) p Fl 2740 1285 a(\)) 2676 1358 y(1) p Fk 2771 1336 a(g) p Fn(;) p 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3513 a(;) p 2508 3513 a(u) p Fk 2591 3513 a(2) p Fn 2685 3513 a(C) p Fg 2762 3472 a(1) p Fl 2755 3537 a(0) p Fo 2837 3513 a(\() p Fj(R) p Fl 2962 3471 a(2) p Fk 3024 3513 a(n) p Fn 3096 3513 a(S) p Fm 3156 3528 a(N) p Fo 3223 3513 a(\)) p Fn(;) p Fo 3653 3513 a(\(5.29\)) 0 3733 y(where) p 282 3733 a(w) m(e) p 425 3733 a(used) p 648 3733 a(an) p 784 3733 a(in) m(tegration) p 1280 3733 a(b) m(y) p 1415 3733 a(parts.) p 1703 3733 a(This) p 1926 3733 a(norm) p 2180 3733 a(is) p 2278 3733 a(p) s(ositiv) m(e) p 2639 3733 a(de\014nite) p 2986 3733 a(since) 649 3953 y(\() p Fk(L) p Fm 756 3968 a(N) p Fn 823 3953 a(u;) p 923 3953 a(u) p Fo(\)) p 1044 3953 a(=) p 1147 3953 a(\(\() p Fk(A) p Fg 1303 3906 a(y) p Fm 1303 3977 a(N) p Fk 1370 3953 a(A) p Fm 1450 3968 a(N) p Fo 1539 3953 a(+) p Fn 1637 3953 a(B) p Fo 1716 3953 a(\)) p Fn(u;) p 1854 3953 a(u) p Fo(\)) p 1974 3953 a(=) p Fk 2078 3953 a(jjA) p Fm 2214 3968 a(N) p Fn 2280 3953 a(u) p Fk(jj) p Fl 2392 3912 a(2) p Fo 2452 3953 a(+) p Fn 2550 3953 a(B) p Fk 2629 3953 a(jj) p Fn(u) p Fk(jj) p Fl 2797 3912 a(2) p Fk 2863 3953 a(\025) p Fn 2968 3953 a(B) p Fk 3047 3953 a(jj) p Fn(u) p Fk(jj) p Fl 3215 3912 a(2) p Fo 3653 3953 a(\(5.30\)) 0 4173 y(b) m(y) p 135 4173 a(Lemma) p 484 4173 a(4.1.) p 679 4173 a(Th) m(us) p Fn 926 4173 a(D) p Fo 1010 4173 a(\() p Fn(H) p Fm 1137 4137 a(AB) 1129 4197 y(N) p Fo 1250 4173 a(\)) p 1320 4173 a(is) p 1418 4173 a(con) m(tained) p 1860 4173 a(in) p 1974 4173 a(the) p 2142 4173 a(Hilb) s(ert) p 2478 4173 a(space) p Fn 961 4393 a(Q) p Fm 1038 4408 a(N) p Fo 1133 4393 a(=) p Fk 1236 4393 a(f) p Fn(u) p Fk 1369 4393 a(2) p Fn 1463 4393 a(L) p Fl 1529 4352 a(2) p Fo 1569 4393 a(\() p Fj(R) p Fl 1694 4351 a(2) p Fo 1734 4393 a(\);) p 1848 4393 a(\() p Fk(r) p Fo 1991 4393 a(+) p Fn 2089 4393 a(i) p Fj(a) p Fm 2185 4408 a(N) p Fo 2253 4393 a(\)) p Fn(u) p Fk 2374 4393 a(2) p Fo 2468 4393 a(\() p Fn(L) p Fl 2572 4352 a(2) p Fo 2612 4393 a(\() p Fj(R) p Fl 2737 4351 a(2) p Fo 2776 4393 a(\)\)) p 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631 a(W) p 238 631 a(e) p 314 631 a(shall) p 542 631 a(sho) m(w) p 784 631 a(that) p Fn 961 856 a(D) p Fo 1045 856 a(\() p Fn(L) p Fg 1149 815 a(\003) p Fm 1149 881 a(N) p Fo 1216 856 a(\)) p Fk 1276 856 a(\\) p Fn 1365 856 a(Q) p Fm 1442 871 a(N) p Fo 1537 856 a(=) p Fn 1640 856 a(D) p Fo 1724 856 a(\() p 1762 778 134 4 v Fn(L) p Fm 1828 871 a(N) p Fo 1896 856 a(\)) p Fk 1956 856 a(\010) p Fm 2033 871 a(al) q(g) 2184 782 y(N) p Fk 2177 856 a(\010) p Fm 2155 922 a(j) p Fl 2188 922 a(=1) p Fo 2294 856 a(L) p Fn(:) p Fo(h) p Fn(:) p Fk(f) p Fn(\036) p Fl 2571 806 a(\() p Fm(j) p Fl 2631 806 a(\)) 2571 878 y(0) p Fn 2662 856 a(;) p 2706 856 a( ) p Fl 2773 806 a(\() p Fm(j) p Fl 2833 806 a(\)) 2769 878 y(1) p Fk 2864 856 a(g) p Fn(:) p Fo 3653 856 a(\(5.33\)) 0 1125 y(Since) p Fn 266 1125 a(D) p Fo 350 1125 a(\() p 388 1047 V Fn(L) p Fm 454 1140 a(N) p Fo 522 1125 a(\)) p Fk 607 1125 a(\032) p Fn 732 1125 a(D) p Fo 816 1125 a(\() p Fn(H) p Fm 943 1089 a(AB) 935 1150 y(N) p Fo 1056 1125 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628 2077 a(\() p Fn(L) p Fl 732 2041 a(2) p Fo 772 2077 a(\() p Fj(R) p Fl 897 2035 a(2) p Fo 937 2077 a(\)\)) p Fl 1013 2041 a(2) p Fo 1086 2077 a(is) p 1186 2077 a(equiv) p 1408 2077 a(alen) m(t) p 1651 2077 a(to) p 1772 2077 a(the) p 1942 2077 a(condition) p Fk 2372 2077 a(A) p Fm 2452 2092 a(N) p Fn 2519 2077 a(u) p Fk 2605 2077 a(2) p Fn 2702 2077 a(L) p Fl 2768 2041 a(2) p Fo 2808 2077 a(\() p Fj(R) p Fl 2933 2035 a(2) p Fo 2973 2077 a(\)) p 3045 2077 a(and) p Fk 3236 2077 a(A) p Fg 3316 2030 a(y) p Fm 3316 2101 a(N) p Fn 3383 2077 a(u) p Fk 3469 2077 a(2) p Fn 3566 2077 a(L) p Fl 3632 2041 a(2) p Fo 3672 2077 a(\() p Fj(R) p Fl 3798 2035 a(2) p Fo 3837 2077 a(\).) 0 2197 y(In) p 122 2197 a(the) p 290 2197 a(op) s(en) p 526 2197 a(set) p Fk 678 2197 a(fj) p Fn(z) p Fk 827 2197 a(\000) p Fn 927 2197 a(z) p Fm 972 2212 a(j) p Fk 1009 2197 a(j) p Fn 1064 2197 a(<) p Fm 1178 2158 a(R) p 1178 2174 54 4 v Fl 1187 2232 a(4) p Fk 1241 2197 a(g) p Fo(,) p 1351 2197 a(w) m(e) p 1494 2197 a(ha) m(v) m(e) p Fk 496 2428 a(A) p Fm 576 2443 a(N) p Fn 643 2428 a(u) p Fo 781 2428 a(=) p Fn 940 2428 a(t) p Fm 975 2443 a(z) p Fe 1008 2453 a(j) p Fk 1045 2428 a(A) p Fm 1125 2374 a(\013) p Fe 1170 2384 a(j) p Fl 1125 2450 a(1) p Fi 1223 2332 a(\020) p Fn 1273 2428 a(c) p Fl 1315 2443 a(4) p Fm(j) p Fg 1383 2443 a(\000) p Fl(3) p Fn 1477 2428 a(\036) p Fm 1535 2374 a(\013) p Fe 1580 2384 a(j) p Fg 1535 2450 a(\000) p Fl(1) p Fm(;) p Fl(0) p Fo 1706 2428 a(+) p Fn 1804 2428 a(c) p Fl 1846 2443 a(4) p Fm(j) p Fg 1914 2443 a(\000) p Fl(2) p Fn 2008 2428 a( ) p Fm 2075 2374 a(\013) p Fe 2120 2384 a(j) p Fl 2071 2450 a(0) p Fm(;) p Fl(1) p Fo 2187 2428 a(+) p Fn 2285 2428 a(c) p Fl 2327 2443 a(4) p Fm(j) p Fg 2395 2443 a(\000) p Fl(1) p Fn 2489 2428 a(\036) p Fm 2547 2374 a(\013) p Fe 2592 2384 a(j) p Fl 2547 2450 a(0) p Fm(;) p Fl(0) p Fo 2664 2428 a(+) p Fn 2762 2428 a(c) p Fl 2804 2443 a(4) p Fm(j) p Fn 2875 2428 a( ) p Fm 2942 2374 a(\013) p Fe 2987 2384 a(j) p Fl 2938 2450 a(0) p Fm(;) p Fl(0) p Fi 3033 2332 a(\021) p Fo 781 2636 a(=) p Fn 940 2636 a(t) p Fm 975 2651 a(z) p Fe 1008 2661 a(j) p Fi 1045 2515 a(\032) p Fo 1107 2636 a(2) p Fn(c) p Fl 1198 2651 a(4) p Fm(j) p Fg 1266 2651 a(\000) p Fl(3) p Fo 1360 2636 a(\() p Fn(\013) p Fm 1460 2651 a(j) p Fk 1518 2636 a(\000) p Fo 1618 2636 a(1\)) p Fn(\036) p Fm 1763 2582 a(\013) p Fe 1808 2592 a(j) p Fg 1763 2658 a(\000) p Fl(2) p Fm(;) p Fl(0) p Fo 1934 2636 a(+) p 2032 2636 a(2) p Fn(c) p Fl 2123 2651 a(4) p Fm(j) p Fg 2191 2651 a(\000) p Fl(1) p Fn 2285 2636 a(\013) p Fm 2347 2651 a(j) p Fn 2383 2636 a(\036) p Fm 2441 2582 a(\013) p Fe 2486 2592 a(j) p Fg 2441 2658 a(\000) p Fl(1) p Fm(;) p Fl(0) p Fi 2590 2515 a(\033) p Fn 2652 2636 a(;) p Fo 3653 2636 a(\(5.35\)) p Fk 496 2844 a(A) p Fg 576 2797 a(y) p Fm 576 2868 a(N) p Fn 643 2844 a(u) p Fo 781 2844 a(=) p Fn 940 2844 a(t) p Fm 975 2859 a(z) p Fe 1008 2869 a(j) p Fk 1045 2844 a(A) p Fg 1125 2790 a(y) p Fm 1156 2790 a(;\013) p Fe 1221 2800 a(j) p Fl 1125 2866 a(1) p Fi 1274 2748 a(\020) p Fn 1324 2844 a(c) p Fl 1366 2859 a(4) p Fm(j) p Fg 1434 2859 a(\000) p Fl(3) p Fn 1528 2844 a(\036) p Fm 1586 2790 a(\013) p Fe 1631 2800 a(j) p Fg 1586 2866 a(\000) p Fl(1) p Fm(;) p Fl(0) p Fo 1757 2844 a(+) p Fn 1855 2844 a(c) p Fl 1897 2859 a(4) p Fm(j) p Fg 1965 2859 a(\000) p Fl(2) p Fn 2059 2844 a( ) p Fm 2126 2790 a(\013) p Fe 2171 2800 a(j) p Fl 2122 2866 a(0) p Fm(;) p Fl(1) p Fo 2238 2844 a(+) p Fn 2336 2844 a(c) p Fl 2378 2859 a(4) p Fm(j) p Fg 2446 2859 a(\000) p Fl(1) p Fn 2540 2844 a(\036) p Fm 2598 2790 a(\013) p Fe 2643 2800 a(j) p Fl 2598 2866 a(0) p Fm(;) p Fl(0) p Fo 2715 2844 a(+) p Fn 2813 2844 a(c) p Fl 2855 2859 a(4) p Fm(j) p Fn 2926 2844 a( ) p Fm 2993 2790 a(\013) p Fe 3038 2800 a(j) p Fl 2989 2866 a(0) p Fm(;) p Fl(0) p Fi 3084 2748 a(\021) p Fo 781 3052 a(=) p Fn 940 3052 a(t) p Fm 975 3067 a(z) p Fe 1008 3077 a(j) p Fi 1045 2930 a(\032) p Fn 1107 3052 a(c) p Fl 1149 3067 a(4) p Fm(j) p Fg 1217 3067 a(\000) p Fl(3) p Fn 1311 3052 a(B) p 1390 3052 a(\036) p Fm 1448 2998 a(\013) p Fe 1493 3008 a(j) p Fl 1448 3073 a(0) p Fm(;) p Fl(0) p Fo 1564 3052 a(+) p Fn 1662 3052 a(c) p Fl 1704 3067 a(4) p Fm(j) p Fg 1772 3067 a(\000) p Fl(2) p Fo 1866 3052 a(\(2\() p Fn(\013) p Fm 2053 3067 a(j) p Fk 2112 3052 a(\000) p Fo 2211 3052 a(1\)) p Fn( ) p Fm 2365 2998 a(\013) p Fe 2410 3008 a(j) p Fl 2361 3073 a(0) p Fm(;) p Fl(0) p Fo 2478 3052 a(+) p Fn 2576 3052 a(B) p 2655 3052 a( ) p Fm 2722 2998 a(\013) p Fe 2767 3008 a(j) p Fl 2718 3073 a(1) p Fm(;) p Fl(1) p Fo 2813 3052 a(\)) p 2873 3052 a(+) p Fn 2971 3052 a(c) p Fl 3013 3067 a(4) p Fm(j) p Fg 3081 3067 a(\000) p Fl(1) p Fn 3175 3052 a(B) p 3254 3052 a(\036) p Fm 3312 2998 a(\013) p Fe 3357 3008 a(j) p Fl 3312 3073 a(1) p Fm(;) p Fl(0) p Fo 1125 3284 a(+) p Fn 1223 3284 a(c) p Fl 1265 3299 a(4) p Fm(j) p Fo 1337 3284 a(\(2) p Fn(\013) p Fm 1486 3299 a(j) p Fn 1522 3284 a( ) p Fm 1589 3230 a(\013) p Fe 1634 3240 a(j) p Fl 1585 3306 a(0) p Fm(;) p Fg(\000) p Fl(1) p Fo 1757 3284 a(+) p Fn 1855 3284 a(B) p 1934 3284 a( ) p Fm 2001 3230 a(\013) p Fe 2046 3240 a(j) p Fl 1997 3306 a(1) p Fm(;) p Fl(0) p Fo 2092 3284 a(\)) p Fn(:) p Fi 2157 3163 a(\033) p Fo 3653 3284 a(\(5.36\)) 0 3541 y(b) m(y) p 137 3541 a(\(ii\)) p 301 3541 a(of) p 413 3541 a(Prop) s(osition) p 940 3541 a(3.1.) p 1139 3541 a(Since) p 1395 3541 a(2\() p Fn(\013) p Fm 1544 3556 a(j) p Fk 1604 3541 a(\000) p Fo 1704 3541 a(1\)) p Fn(\036) p Fm 1849 3486 a(\013) p Fe 1894 3496 a(j) p Fg 1849 3562 a(\000) p Fl(2) p Fm(;) p Fl(0) p Fn 2040 3541 a(=) p Fk 2028 3541 a(2) p Fn 2124 3541 a(L) p Fl 2190 3504 a(2) p Fo 2230 3541 a(\() p Fk(fj) p Fn(z) p Fk 2395 3541 a(j) p Fn 2453 3541 a(<) p Fm 2569 3501 a(R) p 2569 3517 V Fl 2578 3575 a(4) p Fk 2633 3541 a(g) p Fo(\)) p 2754 3541 a(and) p Fn 2945 3541 a(\036) p Fm 3003 3486 a(\013) p Fe 3048 3496 a(j) p Fg 3003 3562 a(\000) p Fl(1) p Fm(;) p Fl(0) p Fk 3182 3541 a(2) p Fn 3279 3541 a(L) p Fl 3345 3504 a(2) p Fo 3385 3541 a(\() p Fk(fj) p Fn(z) p Fk 3550 3541 a(j) p Fn 3608 3541 a(<) p Fm 3724 3501 a(R) p 3724 3517 V Fl 3733 3575 a(4) p Fk 3787 3541 a(g) p Fo(\),) 0 3661 y(w) m(e) p 143 3661 a(see) p 300 3661 a(that) p Fk 510 3661 a(A) p Fm 590 3676 a(N) p Fn 657 3661 a(u) p Fk 740 3661 a(2) p Fn 834 3661 a(L) p Fl 900 3625 a(2) p Fo 940 3661 a(\() p Fj(R) p Fl 1065 3619 a(2) p Fo 1105 3661 a(\)) p 1174 3661 a(implies) p Fn 1505 3661 a(c) p Fl 1547 3676 a(4) p Fm(j) p Fg 1615 3676 a(\000) p Fl(3) p Fo 1736 3661 a(=) p 1840 3661 a(0.) p 1959 3661 a(Since) p 2213 3661 a(2) p Fn(\013) p Fm 2324 3676 a(j) p Fn 2360 3661 a( ) p Fl 2423 3676 a(0) p Fm(;) p Fg(\000) p Fl(1) p Fn 2612 3661 a(=) p Fk 2600 3661 a(2) p Fn 2694 3661 a(L) p Fl 2760 3625 a(2) p Fo 2800 3661 a(\() p Fk(fj) p Fn(z) p Fk 2965 3661 a(j) p Fn 3021 3661 a(<) p Fm 3134 3622 a(R) p 3134 3638 V Fl 3143 3695 a(4) p Fk 3198 3661 a(g) p Fo(\)) p 3317 3661 a(and) p Fn 3506 3661 a(\036) p Fm 3564 3607 a(\013) p Fe 3609 3617 a(j) p Fl 3564 3683 a(0) p Fm(;) p Fl(0) p Fo 3658 3661 a(,) p Fn 3717 3661 a( ) p Fm 3784 3607 a(\013) p Fe 3829 3617 a(j) p Fl 3780 3683 a(0) p Fm(;) p Fl(0) p Fo 3875 3661 a(,) p Fn 0 3797 a( ) p Fm 67 3743 a(\013) p Fe 112 3753 a(j) p Fl 63 3819 a(1) p Fm(;) p Fl(1) p Fo 158 3797 a(,) p Fn 217 3797 a(\036) p Fm 275 3743 a(\013) p Fe 320 3753 a(j) p Fl 275 3819 a(1) p Fm(;) p Fl(0) p Fo 369 3797 a(,) p Fn 429 3797 a( ) p Fm 496 3743 a(\013) p Fe 541 3753 a(j) p Fl 492 3819 a(1) p Fm(;) p Fl(0) p Fk 614 3797 a(2) p Fn 709 3797 a(L) p Fl 775 3761 a(2) p Fo 814 3797 a(\() p Fk(fj) p Fn(z) p Fk 979 3797 a(j) p Fn 1035 3797 a(<) p Fm 1148 3758 a(R) p 1148 3774 V Fl 1157 3832 a(4) p Fk 1212 3797 a(g) p Fo(\),) p 1359 3797 a(w) m(e) p 1503 3797 a(see) p 1661 3797 a(that) p Fk 1872 3797 a(A) p Fg 1952 3750 a(y) p Fm 1952 3822 a(N) p Fn 2019 3797 a(u) p Fk 2102 3797 a(2) p Fn 2196 3797 a(L) p Fl 2262 3761 a(2) p Fo 2302 3797 a(\() p Fj(R) p Fl 2427 3755 a(2) p Fo 2467 3797 a(\)) p 2537 3797 a(implies) p Fn 2868 3797 a(c) p Fl 2910 3812 a(4) p Fm(j) p Fo 3010 3797 a(=) p 3113 3797 a(0.) p 3232 3797 a(Th) m(us) p 3479 3797 a(w) m(e) p 3623 3797 a(obtain) p Fn 960 4057 a(D) p Fo 1044 4057 a(\() p Fn(L) p Fg 1148 4016 a(\003) p Fm 1148 4082 a(N) p Fo 1215 4057 a(\)) p Fk 1275 4057 a(\\) p Fn 1364 4057 a(Q) p Fm 1441 4072 a(N) p Fk 1536 4057 a(\032) p Fn 1641 4057 a(D) p Fo 1725 4057 a(\() p 1763 3979 134 4 v Fn(L) p Fm 1829 4072 a(N) p Fo 1896 4057 a(\)) p Fk 1957 4057 a(\010) p Fm 2034 4072 a(al) q(g) 2185 3983 y(N) p Fk 2178 4057 a(\010) p Fm 2156 4123 a(j) p Fl 2189 4123 a(=1) p Fo 2295 4057 a(L) p Fn(:) p Fo(h) p Fn(:) p Fk(f) p Fn(\036) p Fl 2572 4006 a(\() p Fm(j) p Fl 2632 4006 a(\)) 2572 4079 y(0) p Fn 2663 4057 a(;) p 2707 4057 a( ) p Fl 2774 4006 a(\() p Fm(j) p Fl 2834 4006 a(\)) 2770 4079 y(1) p Fk 2865 4057 a(g) p Fn(:) p Fo 0 4342 a(Con) m(v) m(ersely) p 451 4342 a(,) p 511 4342 a(tak) m(e) p Fn 721 4342 a(u) p Fo 804 4342 a(=) p Fi 908 4276 a(P) p Fm 996 4302 a(N) 996 4367 y(j) p Fl 1029 4367 a(=1) p Fo 1122 4342 a(\() p Fn(c) p Fl 1202 4357 a(4) p Fm(j) p Fg 1270 4357 a(\000) p Fl(2) p Fn 1364 4342 a( ) p Fl 1431 4292 a(\() p Fm(j) p Fl 1491 4292 a(\)) 1427 4364 y(1) p Fo 1541 4342 a(+) p Fn 1636 4342 a(c) p Fl 1678 4357 a(4) p Fm(j) p Fg 1746 4357 a(\000) p Fl(1) p Fn 1840 4342 a(\036) p Fl 1898 4292 a(\() p Fm(j) p Fl 1958 4292 a(\)) 1898 4364 y(0) p Fo 1989 4342 a(\)) p Fk 2055 4342 a(2) p 2149 4342 a(\010) p Fm 2227 4306 a(N) 2227 4367 y(j) p Fl 2260 4367 a(=1) p Fo 2370 4342 a(L) p Fn(:) p Fo(h) p Fn(:) p Fk(f) p Fn(\036) p Fl 2647 4292 a(\() p Fm(j) p Fl 2707 4292 a(\)) 2647 4364 y(0) p Fn 2738 4342 a(;) p 2782 4342 a( ) p Fl 2849 4292 a(\() p Fm(j) p Fl 2909 4292 a(\)) 2845 4364 y(1) p Fk 2940 4342 a(g) p Fo(.) p 3060 4342 a(By) p 3211 4342 a(the) p 3378 4342 a(ab) s(o) m(v) m(e) p 3653 4342 a(calcu-) 0 4463 y(lus,) p 182 4463 a(the) p 352 4463 a(Leibniz) p 697 4463 a(rule) p 894 4463 a(and) p 1086 4463 a(\(5.11\),) p 1398 4463 a(w) m(e) p 1544 4463 a(ha) m(v) m(e) p 1771 4463 a(that) p 1984 4463 a(there) p 2236 4463 a(exist) p 2469 4463 a(a) p 2552 4463 a(constan) m(t) p Fn 2948 4463 a(C) p 3056 4463 a(>) p Fo 3163 4463 a(0) p 3247 4463 a(dep) s(enden) m(t) p 3720 4463 a(only) 0 4583 y(on) p Fn 135 4583 a(B) p Fo 214 4583 a(,) p Fn 274 4583 a(R) p Fo 349 4583 a(,) p Fn 409 4583 a(\037) p Fo(,) p Fn 530 4583 a(\013) p Fg 592 4598 a(\000) p Fo 651 4583 a(,) p Fn 710 4583 a(\013) p Fl 772 4598 a(+) p Fo 864 4583 a(suc) m(h) p 1084 4583 a(that) p Fk 768 4803 a(jjA) p Fm 904 4818 a(N) p Fo 970 4803 a(\() p Fn(c) p Fl 1050 4818 a(4) p Fm(j) p Fg 1118 4818 a(\000) p Fl(2) p Fn 1212 4803 a( ) p Fl 1279 4752 a(\() p Fm(j) p Fl 1339 4752 a(\)) 1275 4825 y(1) p Fo 1393 4803 a(+) p Fn 1491 4803 a(c) p Fl 1533 4818 a(4) p Fm(j) p Fg 1601 4818 a(\000) p Fl(1) p Fn 1694 4803 a(\036) p Fl 1752 4752 a(\() p Fm(j) p Fl 1812 4752 a(\)) 1752 4825 y(0) p Fo 1844 4803 a(\)) p Fk(jj) p Fl 1938 4762 a(2) p Fk 2059 4803 a(\024) p Fn 2220 4803 a(C) p Fo 2297 4803 a(\() p Fk(j) p Fn(c) p Fl 2405 4818 a(4) p Fm(j) p Fg 2473 4818 a(\000) p Fl(2) p Fk 2566 4803 a(j) p Fl 2594 4762 a(2) p Fo 2656 4803 a(+) p Fk 2754 4803 a(j) p Fn(c) p Fl 2824 4818 a(4) p Fm(j) p Fg 2892 4818 a(\000) p Fl(1) p Fk 2985 4803 a(j) p Fl 3013 4762 a(2) p Fo 3052 4803 a(\)) p Fn(;) p Fo 3653 4803 a(\(5.37\)) p Fk 768 4965 a(jjA) p Fg 904 4917 a(y) p Fm 904 4989 a(N) p Fo 970 4965 a(\() p Fn(c) p Fl 1050 4980 a(4) p Fm(j) p Fg 1118 4980 a(\000) p Fl(2) p Fn 1212 4965 a( ) p Fl 1279 4914 a(\() p Fm(j) p Fl 1339 4914 a(\)) 1275 4986 y(1) p Fo 1393 4965 a(+) p Fn 1491 4965 a(c) p Fl 1533 4980 a(4) p Fm(j) p Fg 1601 4980 a(\000) p Fl(1) p Fn 1694 4965 a(\036) p Fl 1752 4914 a(\() p Fm(j) p Fl 1812 4914 a(\)) 1752 4986 y(0) p Fo 1844 4965 a(\)) p Fk(jj) p Fl 1938 4924 a(2) p Fk 2059 4965 a(\024) p Fn 2220 4965 a(C) p Fo 2297 4965 a(\() p Fk(j) p Fn(c) p Fl 2405 4980 a(4) p Fm(j) p Fg 2473 4980 a(\000) p Fl(2) p Fk 2566 4965 a(j) p Fl 2594 4924 a(2) p Fo 2656 4965 a(+) p Fk 2754 4965 a(j) p Fn(c) p Fl 2824 4980 a(4) p Fm(j) p Fg 2892 4980 a(\000) p Fl(1) p Fk 2985 4965 a(j) p Fl 3013 4924 a(2) p Fo 3052 4965 a(\)) p Fn(:) p Fo 3653 4965 a(\(5.38\)) 1902 5214 y(37) p eop %%Page: 38 38 38 37 bop Fo 0 632 a(Since) p Fk 255 632 a(f) p Fo(supp) q(\() p Fn(c) p Fl 586 647 a(4) p Fm(j) p Fg 654 647 a(\000) p Fl(2) p Fn 747 632 a( ) p Fl 814 581 a(\() p Fm(j) p Fl 874 581 a(\)) 810 654 y(1) p Fo 928 632 a(+) p Fn 1026 632 a(c) p Fl 1068 647 a(4) p Fm(j) p Fg 1136 647 a(\000) p Fl(1) p Fn 1230 632 a(\036) p Fl 1288 581 a(\() p Fm(j) p Fl 1348 581 a(\)) 1288 654 y(0) p Fm(;) p Fl(0) p Fo 1382 632 a(\)) p Fk(g) p Fm 1470 596 a(N) 1470 657 y(j) p Fl 1503 657 a(=1) p Fo 1629 632 a(are) p 1792 632 a(disjoin) m(t,) p 2166 632 a(w) m(e) p 2310 632 a(ha) m(v) m(e) p Fk 1399 911 a(jjA) p Fm 1535 926 a(N) 1647 803 y(N) p Fi 1619 828 a(X) p Fm 1618 1011 a(j) p Fl 1651 1011 a(=1) p Fo 1740 911 a(\() p Fn(c) p Fl 1820 926 a(4) p Fm(j) p Fg 1888 926 a(\000) p Fl(2) p Fn 1982 911 a( ) p Fl 2049 860 a(\() p Fm(j) p Fl 2109 860 a(\)) 2045 933 y(1) p Fo 2163 911 a(+) p Fn 2261 911 a(c) p Fl 2303 926 a(4) p Fm(j) p Fg 2371 926 a(\000) p Fl(1) p Fn 2465 911 a(\036) p Fl 2523 860 a(\() p Fm(j) p Fl 2583 860 a(\)) 2523 933 y(0) p Fo 2614 911 a(\)) p Fk(jj) p Fl 2708 870 a(2) p Fo 1239 1227 a(=) p Fm 1429 1119 a(N) p Fi 1400 1144 a(X) p Fm 1399 1326 a(j) p Fl 1432 1326 a(=1) p Fk 1538 1227 a(jjA) p Fm 1674 1242 a(N) p Fo 1740 1227 a(\() p Fn(c) p Fl 1820 1242 a(4) p Fm(j) p Fg 1888 1242 a(\000) p Fl(2) p Fn 1982 1227 a( ) p Fl 2049 1176 a(\() p Fm(j) p Fl 2109 1176 a(\)) 2045 1248 y(1) p Fo 2163 1227 a(+) p Fn 2261 1227 a(c) p Fl 2303 1242 a(4) p Fm(j) p Fg 2371 1242 a(\000) p Fl(1) p Fn 2465 1227 a(\036) p Fl 2523 1176 a(\() p Fm(j) p Fl 2583 1176 a(\)) 2523 1248 y(0) p Fo 2614 1227 a(\)) p Fk(jj) p Fl 2708 1185 a(2) p Fk 1238 1542 a(\024) p Fn 1399 1542 a(C) p Fm 1522 1434 a(N) p Fi 1494 1459 a(X) p Fm 1493 1641 a(j) p Fl 1526 1641 a(=1) p Fo 1615 1542 a(\() p Fk(j) p Fn(c) p Fl 1723 1557 a(4) p Fm(j) p Fg 1791 1557 a(\000) p Fl(2) p Fk 1885 1542 a(j) p Fl 1913 1501 a(2) p Fo 1974 1542 a(+) p Fk 2072 1542 a(j) p Fn(c) p Fl 2142 1557 a(4) p Fm(j) p Fg 2210 1557 a(\000) p Fl(1) p Fk 2304 1542 a(j) p Fl 2332 1501 a(2) p Fo 2371 1542 a(\)) p Fn 2436 1542 a(<) p Fk 2540 1542 a(1) p Fo 0 1836 a(b) m(y) p 141 1836 a(\(5.37\).) p 477 1836 a(Th) m(us) p Fk 729 1836 a(A) p Fm 809 1851 a(N) p Fn 876 1836 a(u) p Fk 968 1836 a(2) p Fn 1072 1836 a(L) p Fl 1138 1800 a(2) p Fo 1177 1836 a(\() p Fj(R) p Fl 1303 1794 a(2) p Fo 1342 1836 a(\).) p 1467 1836 a(Similarly) p 1845 1836 a(,) p Fk 1907 1836 a(A) p Fg 1987 1788 a(y) p Fm 1987 1860 a(N) p Fn 2054 1836 a(u) p Fk 2146 1836 a(2) p Fn 2249 1836 a(L) p Fl 2315 1800 a(2) p Fo 2355 1836 a(\() p Fj(R) p Fl 2480 1794 a(2) p Fo 2520 1836 a(\)) p 2596 1836 a(follo) m(ws) p 2921 1836 a(from) p 3157 1836 a(\(5.38\).) p 3493 1836 a(Therefore) 0 1956 y(\(5.33\)) p 282 1956 a(holds.) 146 2076 y(De\014ne) p 448 2076 a(a) p 530 2076 a(b) s(ounded) p 928 2076 a(linear) p 1199 2076 a(op) s(erator) p Fn 1592 2076 a(M) p Fm 1696 2040 a(AB) 1686 2101 y(N) p Fo 1843 2076 a(on) p Fj 1978 2076 a(C) p Fl 2067 2034 a(4) p Fm(N) p Fo 2202 2076 a(b) m(y) p 2337 2076 a(the) p 2505 2076 a(\(\014nite) p 2792 2076 a(or) p 2912 2076 a(in\014nite\)) p 3280 2076 a(matrix) p Fn 541 2447 a(M) p Fm 645 2406 a(AB) 635 2472 y(N) p Fo 787 2447 a(=) p Fi 891 2201 a(0) 891 2347 y(B) 891 2397 y(B) 891 2447 y(B) 891 2500 y(@) p Fn 1005 2266 a(M) p Fm 1109 2230 a(AB) p Fn 1377 2266 a(O) p 1607 2266 a(:) p 1651 2266 a(:) p 1695 2266 a(:) p 1875 2266 a(O) 1075 2386 y(O) p 1306 2386 a(M) p Fm 1410 2350 a(AB) p Fn 1607 2386 a(:) p 1651 2386 a(:) p 1695 2386 a(:) p 1875 2386 a(O) p Fo 1028 2506 a(.) p 1071 2506 a(.) p 1114 2506 a(.) p 1157 2506 a(.) p 1200 2506 a(.) p 1243 2506 a(.) p 1286 2506 a(.) p 1329 2506 a(.) p 1372 2506 a(.) p 1415 2506 a(.) p 1458 2506 a(.) p 1501 2506 a(.) p 1543 2506 a(.) p 1586 2506 a(.) p 1629 2506 a(.) p 1672 2506 a(.) p 1715 2506 a(.) p 1758 2506 a(.) p 1801 2506 a(.) p 1844 2506 a(.) p 1887 2506 a(.) p 1930 2506 a(.) p 1973 2506 a(.) p Fn 1075 2627 a(O) p 1377 2627 a(O) p 1607 2627 a(:) p 1651 2627 a(:) p 1695 2627 a(:) p 1805 2627 a(M) p Fm 1909 2591 a(AB) p Fi 2065 2201 a(1) 2065 2347 y(C) 2065 2397 y(C) 2065 2447 y(C) 2065 2500 y(A) p Fn 2154 2447 a(;) p 2295 2447 a(M) p Fm 2399 2406 a(AB) p Fo 2541 2447 a(=) p Fi 2645 2201 a(0) 2645 2347 y(B) 2645 2397 y(B) 2645 2447 y(B) 2645 2500 y(@) p Fo 2759 2266 a(0) p 2891 2266 a(0) p 3022 2266 a(0) p 3154 2266 a(0) 2759 2386 y(0) p 2891 2386 a(1) p 3022 2386 a(0) p 3154 2386 a(0) 2759 2506 y(0) p 2891 2506 a(0) p 3022 2506 a(1) p 3154 2506 a(0) 2759 2627 y(0) p 2891 2627 a(0) p 3022 2627 a(0) p 3154 2627 a(0) p Fi 3245 2201 a(1) 3245 2347 y(C) 3245 2397 y(C) 3245 2447 y(C) 3245 2500 y(A) p Fn 3334 2447 a(:) p Fo 0 2824 a(Then) p 244 2824 a(w) m(e) p 377 2824 a(ha) m(v) m(e) p 591 2824 a(that) p Fn 791 2824 a(M) p Fm 895 2788 a(AB) p Fo 1031 2824 a(is) p 1118 2824 a(a) p 1189 2824 a(orthogonal) p 1666 2824 a(pro) p 1812 2824 a(jection,) p 2148 2824 a(that) p 2348 2824 a(rank) p Fn 2541 2824 a(M) p Fm 2645 2788 a(AB) p Fo 2786 2824 a(=) p 2890 2824 a(2) p 2960 2824 a(and) p 3139 2824 a(that) p Fn 3340 2824 a(M) p Fm 3444 2788 a(AB) p Fn 3558 2824 a(J) p Fm 3612 2839 a(\013) p Fe 3657 2849 a(j) p Fn 3694 2824 a(M) p Fm 3798 2788 a(AB) p Fo 3940 2824 a(=) p Fn 0 2961 a(O) p Fo 78 2961 a(.) p 146 2961 a(By) p 293 2961 a(\(ii\)) p 450 2961 a(of) p 555 2961 a(Lemma) p 897 2961 a(5.6,) p Fn 1077 2961 a(M) p Fm 1181 2925 a(AB) p Fo 1322 2961 a(satis\014es) p 1675 2961 a(\(5.18\).) p 1993 2961 a(By) p 2140 2961 a(\(i\)) p 2270 2961 a(of) p 2375 2961 a(Prop) s(osition) p 2894 2961 a(5.4,) p 3073 2961 a(w) m(e) p 3211 2961 a(ha) m(v) m(e) p 3430 2961 a(that) p Fn 3636 2961 a(H) p Fm 3725 2925 a(M) p Fe 3800 2902 a(AB) p Fm 3717 2986 a(N) p Fo 0 3082 a(is) p 98 3082 a(a) p 179 3082 a(self-adjoin) m(t) p 681 3082 a(extension) p 1113 3082 a(of) p Fn 1224 3082 a(L) p Fm 1290 3097 a(N) p Fo 1390 3082 a(and) p Fn 1022 3309 a(D) p Fo 1106 3309 a(\() p Fn(H) p Fm 1233 3268 a(M) p Fe 1308 3245 a(AB) p Fm 1225 3334 a(N) p Fo 1410 3309 a(\)) p 1476 3309 a(=) p Fn 1579 3309 a(D) p Fo 1663 3309 a(\() p 1701 3231 134 4 v Fn(L) p Fm 1767 3324 a(N) p Fo 1834 3309 a(\)) p Fk 1894 3309 a(\010) p Fm 1971 3324 a(al) q(g) 2123 3235 y(N) p Fk 2116 3309 a(\010) p Fm 2093 3375 a(j) p Fl 2126 3375 a(=1) p Fo 2233 3309 a(L) p Fn(:) p Fo(h) p Fn(:) p Fk(f) p Fn(\036) p Fl 2510 3258 a(\() p Fm(j) p Fl 2570 3258 a(\)) 2510 3331 y(0) p Fn 2601 3309 a(;) p 2645 3309 a( ) p Fl 2712 3258 a(\() p Fm(j) p Fl 2772 3258 a(\)) 2708 3331 y(1) p Fk 2803 3309 a(g) p Fn(:) p Fo 3653 3309 a(\(5.39\)) 0 3570 y(This) p 225 3570 a(equalit) m(y) p 556 3570 a(,) p 619 3570 a(\(5.32\)) p 903 3570 a(and) p 1096 3570 a(\(5.33\)) p 1380 3570 a(implies) p 1714 3570 a(the) p 1884 3570 a(op) s(erator) p 2280 3570 a(inclusion) p Fn 2690 3570 a(H) p Fm 2779 3534 a(AB) 2771 3595 y(N) p Fk 2924 3570 a(\032) p Fn 3033 3570 a(H) p Fm 3122 3534 a(M) p Fe 3197 3511 a(AB) p Fm 3114 3595 a(N) p Fo 3300 3570 a(.) p 3378 3570 a(This) p 3603 3570 a(implies) p Fn 0 3691 a(H) p Fm 89 3655 a(AB) 81 3715 y(N) p Fo 237 3691 a(=) p Fn 348 3691 a(H) p Fm 437 3655 a(M) p Fe 512 3631 a(AB) p Fm 429 3715 a(N) p Fo 615 3691 a(,) p 680 3691 a(since) p 923 3691 a(b) s(oth) p 1158 3691 a(op) s(erators) p 1594 3691 a(are) p 1761 3691 a(self-adjoin) m(t.) p 2314 3691 a(Th) m(us) p 2565 3691 a(\(i\)) p 2705 3691 a(and) p 2899 3691 a(\(ii\)) p 3066 3691 a(follo) m(ws) p 3391 3691 a(from) p 3626 3691 a(\(5.32\),) 0 3811 y(\(5.33\)) p 282 3811 a(and) p 471 3811 a(\(5.39\).) 146 3931 y(\(iii\)) p 336 3931 a(These) p 618 3931 a(inequalities) p 1131 3931 a(follo) m(ws) p 1451 3931 a(from) p 1682 3931 a(\(5.29\),) p 1991 3931 a(\(5.30\)) p 2272 3931 a(and) p 2462 3931 a(an) p 2598 3931 a(appro) m(ximation) p 3248 3931 a(argumen) m(t.) 146 4052 y(\(iv\)) p 334 4052 a(F) p 390 4052 a(or) p Fn 509 4052 a(u) p Fk 593 4052 a(2) p Fn 688 4052 a(D) p Fo 772 4052 a(\() p Fn(H) p Fm 899 4016 a(AB) 891 4076 y(N) p Fo 1012 4052 a(\)) p 1083 4052 a(and) p Fn 1273 4052 a(\037) p Fk 1363 4052 a(2) p Fn 1458 4052 a(C) p Fg 1535 4016 a(1) p Fo 1610 4052 a(\() p Fj(R) p Fl 1735 4010 a(2) p Fo 1774 4052 a(\)) p 1845 4052 a(with) p Fn 2068 4052 a(\037) p Fo(,) p Fk 2189 4052 a(r) p Fn(\037) p Fo 2367 4052 a(and) p 2557 4052 a(\001) p Fn(\037) p Fo 2732 4052 a(b) s(ounded,) p 3158 4052 a(the) p 3327 4052 a(distributional) 0 4172 y(equalit) m(y) 608 4293 y(\() p Fk(r) p Fo 751 4293 a(+) p Fn 849 4293 a(i) p Fj(a) p Fm 946 4308 a(N) p Fo 1013 4293 a(\)) p Fl 1051 4251 a(2) p Fo 1090 4293 a(\() p Fn(\037u) p Fo(\)) p 1310 4293 a(=) p Fn 1414 4293 a(\037) p Fo(\() p Fk(r) p Fo 1618 4293 a(+) p Fn 1716 4293 a(i) p Fj(a) p Fm 1813 4308 a(N) p Fo 1880 4293 a(\)) p Fl 1918 4251 a(2) p Fn 1957 4293 a(u) p Fo 2035 4293 a(+) p 2133 4293 a(2\() p Fk(r) p Fn(\037) p Fo(\)\() p Fk(r) p Fo 2545 4293 a(+) p Fn 2643 4293 a(i) p Fj(a) p Fm 2739 4308 a(N) p Fo 2806 4293 a(\)) p Fn(u) p Fo 2922 4293 a(+) p 3020 4293 a(\(\001) p Fn(\037) p Fo(\)) p Fn(u) p Fo 3653 4293 a(\(5.40\)) 0 4455 y(holds) p 259 4455 a(in) p Fn 377 4455 a(D) p Fg 461 4419 a(0) p Fo 484 4455 a(\() p Fj(R) p Fl 609 4413 a(2) p Fk 673 4455 a(n) p Fn 748 4455 a(S) p Fm 808 4470 a(N) p Fo 875 4455 a(\).) p 995 4455 a(By) p 1152 4455 a(\(ii\)) p 1318 4455 a(and) p 1511 4455 a(the) p 1683 4455 a(assumption,) p 2231 4455 a(the) p 2403 4455 a(righ) m(t) p 2642 4455 a(hand) p 2890 4455 a(side) p 3090 4455 a(b) s(elongs) p 3443 4455 a(to) p Fn 3566 4455 a(L) p Fl 3632 4419 a(2) p Fo 3672 4455 a(\() p Fj(R) p Fl 3798 4413 a(2) p Fo 3837 4455 a(\).) 0 4575 y(Th) m(us) p Fn 247 4575 a(\037u) p Fk 391 4575 a(2) p Fn 485 4575 a(D) p Fo 569 4575 a(\() p Fn(L) p Fg 673 4539 a(\003) p Fm 673 4600 a(N) p Fo 740 4575 a(\)) p 811 4575 a(b) m(y) p 946 4575 a(\(i\)) p 1081 4575 a(of) p 1192 4575 a(Prop) s(osition) p 1716 4575 a(4.2.) p 1911 4575 a(One) p 2117 4575 a(can) p 2296 4575 a(sho) m(w) p 2537 4575 a(that) p 2748 4575 a(the) p 2916 4575 a(m) m(ultiplication) p 3542 4575 a(op) s(erator) 0 4696 y(b) m(y) p Fn 149 4696 a(\037) p Fo 256 4696 a(is) p 368 4696 a(a) p 462 4696 a(b) s(ounded) p 874 4696 a(op) s(erator) p 1281 4696 a(from) p Fn 1524 4696 a(Q) p Fm 1601 4711 a(N) p Fo 1715 4696 a(to) p Fn 1847 4696 a(Q) p Fm 1924 4711 a(N) p Fo 1992 4696 a(,) p 2068 4696 a(b) m(y) p 2217 4696 a(the) p 2399 4696 a(Leibniz) p 2755 4696 a(rule.) p 3029 4696 a(Th) m(us) p Fn 3289 4696 a(\037u) p Fo 3452 4696 a(b) s(elongs) p 3815 4696 a(to) p Fn 0 4816 a(D) p Fo 84 4816 a(\() p Fn(L) p Fg 188 4780 a(\003) p Fm 188 4841 a(N) p Fo 255 4816 a(\)) p Fk 322 4816 a(\\) p Fn 417 4816 a(Q) p Fm 494 4831 a(N) p Fo 562 4816 a(,) p 634 4816 a(whic) m(h) p 923 4816 a(equals) p 1228 4816 a(to) p Fn 1357 4816 a(D) p Fo 1441 4816 a(\() p Fn(H) p Fm 1568 4780 a(AB) 1560 4841 y(N) p Fo 1681 4816 a(\)) p 1761 4816 a(b) m(y) p 1907 4816 a(\(ii\)) p 2079 4816 a(of) p 2200 4816 a(this) p 2400 4816 a(prop) s(osition.) p 2980 4816 a(The) p 3190 4816 a(inequalit) m(y) p 3653 4816 a(\(5.28\)) 0 4936 y(follo) m(ws) p 320 4936 a(from) p 551 4936 a(\(5.40\)) p 832 4936 a(and) p 1022 4936 a(an) p 1158 4936 a(appro) m(ximating) p 1808 4936 a(argumen) m(t.) p 2282 4869 64 4 v 2282 4961 4 93 v 2342 4961 V 2282 4964 64 4 v 1902 5214 a(38) p eop %%Page: 39 39 39 38 bop Fh 0 631 a(6) p 242 631 a(P) l(erturb) t(ed) p 1017 631 a(canonical) p 1721 631 a(comm) l(utation) p 2700 631 a(relation) p Fo 0 850 a(In) p 121 850 a(this) p 310 850 a(section,) p 661 850 a(w) m(e) p 804 850 a(de\014ne) p 1084 850 a(the) p 1251 850 a(annihilation) p 1792 850 a(op) s(erators) p 2222 850 a(and) p 2410 850 a(the) p 2577 850 a(creation) p 2949 850 a(op) s(erators,) p 3407 850 a(and) p 3595 850 a(in) m(v) m(esti-) 0 970 y(gate) p 211 970 a(their) p 444 970 a(comm) m(utation) p 1038 970 a(relations) p 1434 970 a(as) p 1554 970 a(linear) p 1825 970 a(op) s(erators) p 2256 970 a(on) p 2392 970 a(the) p 2560 970 a(Hilb) s(ert) p 2896 970 a(space) p Fn 3156 970 a(L) p Fl 3222 934 a(2) p Fo 3262 970 a(\() p Fj(R) p Fl 3387 928 a(2) p Fo 3427 970 a(\).) p Fq 146 1169 a(Lemma) p 543 1169 a(6.1) p Ff 735 1169 a(De\014ne) p 1043 1169 a(line) p 1193 1169 a(ar) p 1318 1169 a(op) p 1413 1169 a(er) p 1494 1169 a(ators) p Fn 1742 1169 a(A) p Fm 1815 1184 a(N) p Ff 1882 1169 a(,) p Fn 1947 1169 a(A) p Fg 2020 1122 a(y) p Fm 2020 1194 a(N) p Ff 2122 1169 a(by) p Fn 1039 1384 a(A) p Fm 1112 1399 a(N) p Fn 1180 1384 a(u) p Fo 1318 1384 a(:=) p Fk 1504 1384 a(A) p Fm 1584 1399 a(N) p Fn 1651 1384 a(u;) p 1785 1384 a(D) p Fo 1869 1384 a(\() p Fn(A) p Fm 1980 1399 a(N) p Fo 2047 1384 a(\)) p 2113 1384 a(:=) p Fn 2243 1384 a(C) p Fg 2320 1343 a(1) p Fl 2313 1409 a(0) p Fo 2395 1384 a(\() p Fj(R) p Fl 2520 1342 a(2) p Fk 2582 1384 a(n) p Fn 2654 1384 a(S) p Fm 2714 1399 a(N) p Fo 2781 1384 a(\)) p Fn(;) 1039 1529 y(A) p Fg 1112 1482 a(y) p Fm 1112 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a(+) p Fn 2169 1864 a(B) p Ff 2283 1864 a(in) p 2402 1864 a(the) p 2564 1864 a(op) p 2659 1864 a(er) p 2740 1864 a(ator) p 2948 1864 a(sense.) 146 1996 y(\(ii\)) p 321 1996 a(We) p 498 1996 a(have) p Fn 722 1996 a(A) p Fg 795 1948 a(y) p Fm 795 2020 a(N) p Fk 890 1996 a(\032) p Fn 995 1996 a(A) p Fm 1068 2011 a(N) p Fg 1135 1954 a(\003) p Ff 1175 1996 a(,) p Fn 1240 1996 a(A) p Fm 1313 2011 a(N) p Fk 1408 1996 a(\032) p Fn 1513 1996 a(A) p Fg 1586 1948 a(y) p Fm 1586 2020 a(N) p Fg 1653 1928 a(\003) p Ff 1727 1996 a(in) p 1847 1996 a(the) p 2009 1996 a(op) p 2104 1996 a(er) p 2185 1996 a(ator) p 2392 1996 a(sense.) p 2691 1996 a(Mor) p 2864 1996 a(e) p 2904 1996 a(over,) p 3150 1996 a(we) p 3294 1996 a(have) p Fn 990 2210 a(D) p Fo 1074 2210 a(\() p Fn(A) p Fm 1185 2225 a(N) p Fg 1252 2168 a(\003) p Fo 1292 2210 a(\)) p 1413 2210 a(=) p Fk 1572 2210 a(f) p Fn(u) p Fk 1705 2210 a(2) p Fn 1799 2210 a(L) p Fl 1865 2169 a(2) p Fo 1904 2210 a(\() p Fj(R) p Fl 2030 2168 a(2) p Fo 2069 2210 a(\);) p Fk 2186 2210 a(A) p Fg 2266 2163 a(y) p Fm 2266 2235 a(N) p Fn 2332 2210 a(u) p Fk 2416 2210 a(2) p Fn 2510 2210 a(L) p Fl 2576 2169 a(2) p Fo 2616 2210 a(\() p Fj(R) p Fl 2741 2168 a(2) p Fo 2780 2210 a(\)) p Fk(g) p Fn(;) 1134 2356 y(A) p Fg 1207 2314 a(\003) p Fm 1207 2380 a(N) p Fn 1274 2356 a(u) p Fo 1413 2356 a(=) p Fk 1572 2356 a(A) p Fg 1652 2308 a(y) p Fm 1652 2380 a(N) p Fn 1718 2356 a(u;) p 1852 2356 a(u) p Fk 1936 2356 a(2) p Fn 2030 2356 a(D) p Fo 2114 2356 a(\() p Fn(A) p Fm 2225 2371 a(N) p Fg 2292 2314 a(\003) p Fo 2331 2356 a(\)) p Fn(;) 990 2512 y(D) p Fo 1074 2512 a(\() p Fn(A) p Fg 1185 2464 a(y) p Fm 1185 2536 a(N) p Fg 1252 2445 a(\003) p Fo 1292 2512 a(\)) p 1413 2512 a(=) p Fk 1572 2512 a(f) p Fn(u) p Fk 1705 2512 a(2) p Fn 1799 2512 a(L) p Fl 1865 2470 a(2) p Fo 1904 2512 a(\() p Fj(R) p Fl 2030 2470 a(2) p Fo 2069 2512 a(\);) p Fk 2186 2512 a(A) p Fm 2266 2527 a(N) p Fn 2332 2512 a(u) p Fk 2416 2512 a(2) p Fn 2510 2512 a(L) p Fl 2576 2470 a(2) p Fo 2616 2512 a(\() p Fj(R) p Fl 2741 2470 a(2) p Fo 2780 2512 a(\)) p Fk(g) p Fn(;) 1094 2668 y(A) p Fg 1167 2620 a(y) p Fm 1167 2692 a(N) p Fg 1235 2601 a(\003) p Fn 1274 2668 a(u) p Fo 1413 2668 a(=) p Fk 1572 2668 a(A) p Fm 1652 2683 a(N) p Fn 1718 2668 a(u;) p 1852 2668 a(u) p Fk 1936 2668 a(2) p Fn 2030 2668 a(D) p Fo 2114 2668 a(\() p Fn(A) p Fg 2225 2620 a(y) p Fm 2225 2692 a(N) p Fg 2292 2601 a(\003) p Fo 2331 2668 a(\)) p Fn(:) p Ff 146 2882 a(\(iii\)) p 351 2882 a(The) p 550 2882 a(op) p 645 2882 a(er) p 726 2882 a(ators) p Fn 973 2882 a(A) p Fm 1046 2897 a(N) p Ff 1148 2882 a(and) p Fn 1338 2882 a(A) p Fg 1411 2835 a(y) p Fm 1411 2907 a(N) p Ff 1513 2882 a(ar) p 1599 2882 a(e) p 1679 2882 a(closable.) p Fo 0 3081 a(Remark) p 367 3081 a(1.) p 486 3081 a(The) p 687 3081 a(pro) s(duct) p Fn 1053 3081 a(S) p 1119 3081 a(T) p Fo 1222 3081 a(of) p 1333 3081 a(t) m(w) m(o) p 1517 3081 a(linear) p 1788 3081 a(op) s(erators) p Fn 2219 3081 a(S) p Fo 2318 3081 a(and) p Fn 2507 3081 a(T) p Fo 2611 3081 a(is) p 2709 3081 a(de\014ned) p 3045 3081 a(b) m(y) p Fn 1132 3296 a(D) p Fo 1216 3296 a(\() p Fn(S) p 1320 3296 a(T) p Fo 1391 3296 a(\)) p 1511 3296 a(=) p Fk 1670 3296 a(f) p Fn(u) p Fk 1803 3296 a(2) p Fn 1897 3296 a(D) p Fo 1981 3296 a(\() p Fn(T) p Fo 2090 3296 a(\);) p Fn 2203 3296 a(T) p 2274 3296 a(u) p Fk 2357 3296 a(2) p Fn 2451 3296 a(D) p Fo 2535 3296 a(\() p Fn(S) p Fo 2639 3296 a(\)) p Fk(g) p Fn(;) 1236 3441 y(S) p 1302 3441 a(T) p 1373 3441 a(u) p Fo 1511 3441 a(=) p Fn 1670 3441 a(S) p Fo 1736 3441 a(\() p Fn(T) p 1845 3441 a(u) p Fo(\)) p Fn(;) p 2079 3441 a(u) p Fk 2162 3441 a(2) p Fn 2256 3441 a(D) p Fo 2340 3441 a(\() p Fn(S) p 2444 3441 a(T) p Fo 2515 3441 a(\)) p Fn(:) p Fo 0 3776 a(Remark) p 372 3776 a(2.) p 506 3776 a(The) p 711 3776 a(op) s(erators) p 1147 3698 141 4 v Fn 1147 3776 a(A) p Fm 1220 3791 a(N) p Fo 1325 3776 a(and) p Fn 1519 3776 a(A) p Fg 1592 3729 a(y) p Fm 1592 3801 a(N) p Fg 1660 3709 a(\003) p Fo 1736 3776 a(are) p 1904 3776 a(di\013eren) m(t.) p 2341 3776 a(The) p 2546 3776 a(op) s(erators) p 2982 3673 V Fn 2982 3776 a(A) p Fg 3055 3729 a(y) p Fm 3055 3801 a(N) p Fo 3160 3776 a(and) p Fn 3354 3776 a(A) p Fm 3427 3791 a(N) p Fg 3495 3735 a(\003) p Fo 3571 3776 a(are) p 3739 3776 a(also) 0 3897 y(di\013eren) m(t.) p Ff 146 4096 a(Pr) p 248 4096 a(o) p 293 4096 a(of.) p Fo 435 4096 a(\(i\)) p 571 4096 a(Since) p Fn 475 4310 a(A) p Fm 548 4325 a(N) p Fn 615 4310 a(C) p Fg 692 4269 a(1) p Fl 685 4335 a(0) p Fo 767 4310 a(\() p Fj(R) p Fl 892 4268 a(2) p Fk 954 4310 a(n) p Fn 1026 4310 a(S) p Fm 1086 4325 a(N) p Fo 1153 4310 a(\)) p Fk 1219 4310 a(\032) p Fn 1324 4310 a(C) p Fg 1401 4269 a(1) p Fl 1394 4335 a(0) p Fo 1475 4310 a(\() p Fj(R) p Fl 1601 4268 a(2) p Fk 1662 4310 a(n) p Fn 1734 4310 a(S) p Fm 1794 4325 a(N) p Fo 1861 4310 a(\)) p Fn(;) p 1976 4310 a(A) p Fg 2049 4263 a(y) p Fm 2049 4335 a(N) p Fn 2116 4310 a(C) p Fg 2193 4269 a(1) p Fl 2186 4335 a(0) p Fo 2268 4310 a(\() p Fj(R) p Fl 2393 4268 a(2) p Fk 2454 4310 a(n) p Fn 2526 4310 a(S) p Fm 2586 4325 a(N) p Fo 2654 4310 a(\)) p Fk 2719 4310 a(\032) p Fn 2824 4310 a(C) p Fg 2901 4269 a(1) p Fl 2894 4335 a(0) p Fo 2976 4310 a(\() p Fj(R) p Fl 3101 4268 a(2) p Fk 3163 4310 a(n) p Fn 3235 4310 a(S) p Fm 3295 4325 a(N) p Fo 3362 4310 a(\)) p Fn(;) p Fo 0 4525 a(w) m(e) p 144 4525 a(ha) m(v) m(e) p Fn 369 4525 a(D) p Fo 453 4525 a(\() p Fn(A) p Fm 564 4540 a(N) p Fn 631 4525 a(A) p Fg 704 4478 a(y) p Fm 704 4549 a(N) p Fo 771 4525 a(\)) p 837 4525 a(=) p Fn 940 4525 a(D) p Fo 1024 4525 a(\() p Fn(A) p Fg 1135 4478 a(y) p Fm 1135 4549 a(N) p Fn 1202 4525 a(A) p Fm 1275 4540 a(N) p Fo 1342 4525 a(\)) p 1408 4525 a(=) p Fn 1512 4525 a(D) p Fo 1596 4525 a(\() p Fn(L) p Fm 1700 4540 a(N) p Fo 1767 4525 a(\).) p 1875 4525 a(Th) m(us) p 2122 4525 a(the) p 2290 4525 a(assertion) p 2698 4525 a(follo) m(ws) p 3018 4525 a(from) p 3248 4525 a(\(4.1\).) 146 4645 y(\(ii\)) p 309 4645 a(This) p 532 4645 a(assertion) p 939 4645 a(follo) m(ws) p 1259 4645 a(from) p 1490 4645 a(\(4.2\)) p 1723 4645 a(and) p 1913 4645 a(the) p 2081 4645 a(de\014nition) p 2514 4645 a(of) p 2625 4645 a(the) p 2793 4645 a(distributional) p 3401 4645 a(deriv) p 3610 4645 a(ativ) m(es.) 146 4766 y(\(iii\)) p 336 4766 a(Since) p Fn 591 4766 a(A) p Fg 664 4730 a(\003) p Fm 664 4790 a(N) p Fo 764 4766 a(and) p 953 4766 a(\() p Fn(A) p Fg 1064 4719 a(y) p Fm 1064 4790 a(N) p Fo 1132 4766 a(\)) p Fg 1170 4730 a(\003) p Fo 1241 4766 a(are) p 1404 4766 a(closed) p 1692 4766 a(op) s(erators,) p 2150 4766 a(this) p 2341 4766 a(assertion) p 2748 4766 a(follo) m(ws) p 3068 4766 a(from) p 3299 4766 a(\(ii\).) p 3499 4698 64 4 v 3499 4791 4 93 v 3559 4791 V 3499 4794 64 4 v 146 4965 a(W) p 238 4965 a(e) p 314 4965 a(review) p 620 4965 a(some) p 865 4965 a(prop) s(erties) p 1323 4965 a(of) p 1434 4965 a(a) p 1516 4965 a(pair) p 1716 4965 a(of) p 1827 4965 a(op) s(erators) p Fn 2259 4965 a(X) p 2348 4965 a(X) p Fg 2437 4929 a(\003) p Fo 2508 4965 a(and) p Fn 2698 4965 a(X) p Fg 2787 4929 a(\003) p Fn 2826 4965 a(X) p Fo 2915 4965 a(.) 1902 5214 y(39) p eop %%Page: 40 40 40 39 bop Fq 146 631 a(Lemma) p 543 631 a(6.2) p Ff 735 631 a(L) p 791 631 a(et) p Fn 901 631 a(X) p Ff 1022 631 a(b) p 1062 631 a(e) p 1139 631 a(a) p 1220 631 a(densely) p 1559 631 a(de\014ne) p 1804 631 a(d) p 1885 631 a(close) p 2085 631 a(d) p 2166 631 a(op) p 2261 631 a(er) p 2342 631 a(ator) p 2547 631 a(on) p 2684 631 a(a) p 2766 631 a(Hilb) p 2933 631 a(ert) p 3083 631 a(sp) p 3168 631 a(ac) p 3258 631 a(e) p Fk 3334 631 a(H) p Ff 3419 631 a(.) p 3493 631 a(Then,) p 3775 631 a(the) 0 751 y(fol) p 110 751 a(lowing) p 413 751 a(holds.) 146 872 y(\(i\)) p 291 872 a(The) p 490 872 a(op) p 585 872 a(er) p 666 872 a(ators) p Fn 913 872 a(X) p Fg 1002 836 a(\003) p Fn 1042 872 a(X) p Ff 1165 872 a(and) p Fn 1355 872 a(X) p 1444 872 a(X) p Fg 1533 836 a(\003) p Ff 1607 872 a(ar) p 1693 872 a(e) p 1772 872 a(self-adjoint.) 146 992 y(\(ii\)) p 311 992 a(The) p 500 992 a(op) p 595 992 a(er) p 676 992 a(ator) p Fo 874 992 a(\() p Fn(X) p 1001 992 a(X) p Fg 1090 956 a(\003) p Fo 1129 992 a(\)) p Fk(j) p Fl 1195 1011 a(\(Ker) p Fm 1347 1011 a(X) p 1410 1011 a(X) p Fb 1473 992 a(\003) p Fl 1509 1011 a(\)) p Fb 1536 992 a(?) p Ff 1618 992 a(is) p 1713 992 a(unitarily) p 2100 992 a(e) p 2140 992 a(quivalent) p 2543 992 a(to) p 2650 992 a(the) p 2802 992 a(op) p 2897 992 a(er) p 2978 992 a(ator) p Fo 3176 992 a(\() p Fn(X) p Fg 3303 956 a(\003) p Fn 3342 992 a(X) p Fo 3431 992 a(\)) p Fk(j) p Fl 3497 1011 a(\(Ker) p Fm 3649 1011 a(X) p Fb 3712 992 a(\003) p Fm 3749 1011 a(X) p Fl 3812 1011 a(\)) p Fb 3839 992 a(?) p Ff 3896 992 a(.) 146 1215 y(Pr) p 248 1215 a(o) p 293 1215 a(of.) p Fo 435 1215 a(\(i\)) p 571 1215 a(See) p 744 1215 a([Re-Si,) p 1060 1215 a(Theorem) p 1472 1215 a(X.25].) 146 1335 y(\(ii\)) p 309 1335 a(See) p 482 1335 a([De,) p 687 1335 a(Theorem) p 1099 1335 a(3].) p 1245 1267 64 4 v 1245 1360 4 93 v 1305 1360 V 1245 1363 64 4 v 146 1584 a(Applying) p 579 1584 a(the) p 758 1584 a(ab) s(o) m(v) m(e) p 1044 1584 a(lemma) p 1369 1584 a(to) p 1499 1584 a(the) p 1677 1584 a(op) s(erators) p 2119 1506 141 4 v Fn 2119 1584 a(A) p Fm 2192 1599 a(N) p Fo 2259 1584 a(,) p 2332 1480 V Fn 2332 1584 a(A) p Fg 2405 1537 a(y) p Fm 2405 1608 a(N) p Fo 2472 1584 a(,) p Fn 2545 1584 a(A) p Fm 2618 1599 a(N) p Fg 2685 1542 a(\003) p Fo 2724 1584 a(,) p Fn 2797 1584 a(A) p Fg 2870 1537 a(y) p Fm 2870 1608 a(N) p Fg 2937 1517 a(\003) p Fo 2977 1584 a(,) p 3049 1584 a(w) m(e) p 3203 1584 a(obtain) p 3517 1584 a(four) p 3731 1584 a(self-) 0 1704 y(adjoin) m(t) p 331 1704 a(op) s(erators.) p 800 1704 a(W) p 892 1704 a(e) p 968 1704 a(shall) p 1196 1704 a(examine) p 1578 1704 a(some) p 1823 1704 a(prop) s(erties) p 2281 1704 a(of) p 2392 1704 a(these) p 2642 1704 a(op) s(erators.) p Fq 146 1903 a(Prop) s(osition) p 752 1903 a(6.3) p Ff 944 1903 a(\(i\)) p Fn 1094 1903 a(A) p Fm 1167 1918 a(N) p Fg 1234 1861 a(\003) p 1273 1825 V Fn 1273 1903 a(A) p Fm 1346 1918 a(N) p Fo 1451 1903 a(=) p Fn 1565 1903 a(H) p Fm 1654 1867 a(AB) 1646 1928 y(N) p Fk 1793 1903 a(\000) p Fn 1896 1903 a(B) p Ff 1975 1903 a(,) p Fn 2047 1903 a(A) p Fg 2120 1856 a(y) p Fm 2120 1928 a(N) p Fg 2187 1836 a(\003) p 2227 1818 109 4 v Fn 2227 1903 a(A) p Fg 2300 1874 a(y) p Fo 2373 1903 a(=) p Fn 2486 1903 a(H) p Fm 2575 1867 a(AB) 2567 1928 y(N) p Fo 2714 1903 a(+) p Fn 2816 1903 a(B) p Ff 2895 1903 a(,) p 2967 1903 a(in) p 3092 1903 a(the) p 3259 1903 a(op) p 3354 1903 a(er) p 3435 1903 a(ator) p 3648 1903 a(sense.) 0 2024 y(Mor) p 173 2024 a(e) p 213 2024 a(over,) p Fn 1175 2256 a(D) p Fo 1259 2256 a(\() p 1297 2178 141 4 v Fn(A) p Fm 1370 2271 a(N) p Fo 1437 2256 a(\)) p Fk 1497 2256 a(\\) p Fn 1586 2256 a(D) p Fo 1670 2256 a(\() p Fn(L) p Fg 1774 2215 a(\003) p Fm 1774 2281 a(N) p Fo 1841 2256 a(\)) p 1907 2256 a(=) p Fn 2010 2256 a(D) p Fo 2094 2256 a(\() p 2132 2153 V Fn(A) p Fg 2205 2209 a(y) p Fm 2205 2280 a(N) p Fo 2272 2256 a(\)) p Fk 2332 2256 a(\\) p Fn 2421 2256 a(D) p Fo 2505 2256 a(\() p Fn(L) p Fg 2609 2215 a(\003) p Fm 2609 2281 a(N) p Fo 2676 2256 a(\)) 1016 2442 y(=) p Fn 1175 2442 a(D) p Fo 1259 2442 a(\() p Fn(H) p Fm 1386 2401 a(AB) 1378 2467 y(N) p Fo 1499 2442 a(\)) p 1564 2442 a(=) p Fn 1668 2442 a(D) p Fo 1752 2442 a(\() p 1790 2364 134 4 v Fn(L) p Fm 1856 2457 a(N) p Fo 1923 2442 a(\)) p Fk 1983 2442 a(\010) p Fm 2060 2457 a(al) q(g) 2212 2368 y(N) p Fk 2205 2442 a(\010) p Fm 2182 2508 a(j) p Fl 2215 2508 a(=1) p Fo 2321 2442 a(L) p Fn(:) p Fo(h) p Fn(:) p Fk 2491 2442 a(f) p Fn(\036) p Fl 2599 2391 a(\() p Fm(j) p Fl 2659 2391 a(\)) 2599 2464 y(0) p Fn 2690 2442 a(;) p 2734 2442 a( ) p Fl 2801 2391 a(\() p Fm(j) p Fl 2861 2391 a(\)) 2797 2464 y(1) p Fk 2892 2442 a(g) p Fn(:) p Fo 3701 2442 a(\(6.1\)) p Ff 146 2717 a(\(ii\)) p 334 2717 a(Ther) p 535 2717 a(e) p 627 2717 a(exists) p 906 2717 a(two) p 1101 2717 a(self-adjoint) p 1620 2717 a(extensions) p Fn 2104 2717 a(H) p Fl 2193 2676 a(+) p Fm 2185 2742 a(N) p Ff 2300 2717 a(and) p Fn 2502 2717 a(H) p Fg 2591 2676 a(\000) p Fm 2583 2742 a(N) p Ff 2698 2717 a(of) p Fn 2826 2717 a(L) p Fm 2892 2732 a(N) p Ff 3007 2717 a(such) p 3242 2717 a(that) p 3454 2639 141 4 v Fn 3454 2717 a(A) p Fm 3527 2732 a(N) p Fn 3595 2717 a(A) p Fm 3668 2732 a(N) p Fg 3735 2675 a(\003) p Fo 3826 2717 a(=) p Fn 0 2860 a(H) p Fl 89 2819 a(+) p Fm 81 2885 a(N) p Fo 170 2860 a(+) p Fn 268 2860 a(B) p Ff 347 2860 a(,) p 412 2757 V Fn 412 2860 a(A) p Fg 485 2813 a(y) p Fm 485 2885 a(N) p Fn 552 2860 a(A) p Fg 625 2813 a(y) p Fm 625 2885 a(N) p Fg 693 2793 a(\003) p Fo 760 2860 a(=) p Fn 863 2860 a(H) p Fg 952 2819 a(\000) p Fm 944 2885 a(N) p Fk 1034 2860 a(\000) p Fn 1133 2860 a(B) p Ff 1247 2860 a(in) p 1367 2860 a(the) p 1529 2860 a(op) p 1624 2860 a(er) p 1705 2860 a(ator) p 1912 2860 a(sense.) 146 2981 y(\(iii\)) p 351 2981 a(The) p 550 2981 a(fol) p 660 2981 a(lowing) p 963 2981 a(de) p 1053 2981 a(c) p 1093 2981 a(omp) p 1268 2981 a(osition) p 1588 2981 a(holds.) p Fn 989 3209 a(D) p Fo 1073 3209 a(\() p Fn(H) p Fl 1200 3168 a(+) p Fm 1192 3234 a(N) p Fo 1258 3209 a(\)) p 1379 3209 a(=) p Fn 1538 3209 a(D) p Fo 1622 3209 a(\() p 1660 3131 134 4 v Fn(L) p Fm 1726 3224 a(N) p Fo 1793 3209 a(\)) p Fk 1854 3209 a(\010) p Fm 1931 3224 a(al) q(g) p Fk 2053 3209 a(\010) p Fm 2130 3168 a(N) 2130 3234 y(j) p Fl 2163 3234 a(=1) p Fo 2257 3209 a(L) p Fn(:) p Fo(h) p Fn(:) p Fk(f) p Fn(\036) p Fl 2534 3158 a(\() p Fm(j) p Fl 2594 3158 a(\)) p Fg 2534 3231 a(\000) p Fl(1) p Fn 2628 3209 a(;) p 2672 3209 a(\036) p Fl 2730 3158 a(\() p Fm(j) p Fl 2790 3158 a(\)) 2730 3231 y(0) p Fk 2821 3209 a(g) p Fo 3701 3209 a(\(6.2\)) p Fn 989 3381 a(D) p Fo 1073 3381 a(\() p Fn(H) p Fg 1200 3340 a(\000) p Fm 1192 3406 a(N) p Fo 1258 3381 a(\)) p 1379 3381 a(=) p Fn 1538 3381 a(D) p Fo 1622 3381 a(\() p 1660 3303 V Fn(L) p Fm 1726 3396 a(N) p Fo 1793 3381 a(\)) p Fk 1854 3381 a(\010) p Fm 1931 3396 a(al) q(g) p Fk 2053 3381 a(\010) p Fm 2130 3340 a(N) 2130 3406 y(j) p Fl 2163 3406 a(=1) p Fo 2257 3381 a(L) p Fn(:) p Fo(h) p Fn(:) p Fk(f) p Fn( ) p Fl 2543 3330 a(\() p Fm(j) p Fl 2603 3330 a(\)) 2539 3403 y(1) p Fn 2634 3381 a(;) p 2678 3381 a( ) p Fl 2745 3330 a(\() p Fm(j) p Fl 2805 3330 a(\)) 2741 3403 y(0) p Fk 2836 3381 a(g) p Fn(:) p Fo 3701 3381 a(\(6.3\)) p Ff 146 3596 a(\(iv\)) p Fo 336 3596 a(dim) p Fn 515 3596 a(D) p Fo 599 3596 a(\() p Fn(H) p Fl 726 3555 a(+) p Fm 718 3620 a(N) p Fo 785 3596 a(\)) p Fn(=) p Fo(\() p Fn(D) p Fo 994 3596 a(\() p Fn(H) p Fm 1121 3560 a(AB) 1113 3621 y(N) p Fo 1233 3596 a(\)) p Fk 1293 3596 a(\\) p Fn 1381 3596 a(D) p Fo 1465 3596 a(\() p Fn(H) p Fl 1592 3555 a(+) p Fm 1584 3620 a(N) p Fo 1651 3596 a(\)\)) p 1755 3596 a(=) p 1858 3596 a(dim) p Fn 2037 3596 a(D) p Fo 2121 3596 a(\() p Fn(H) p Fg 2248 3555 a(\000) p Fm 2240 3620 a(N) p Fo 2307 3596 a(\)) p Fn(=) p Fo(\() p Fn(D) p Fo 2516 3596 a(\() p Fn(H) p Fm 2643 3560 a(AB) 2635 3621 y(N) p Fo 2755 3596 a(\)) p Fk 2816 3596 a(\\) p Fn 2904 3596 a(D) p Fo 2988 3596 a(\() p Fn(H) p Fg 3115 3555 a(\000) p Fm 3107 3620 a(N) p Fo 3174 3596 a(\)\)) p 3277 3596 a(=) p Fn 3381 3596 a(N) p Ff 3469 3596 a(.) 146 3841 y(Pr) p 248 3841 a(o) p 293 3841 a(of.) p Fo 429 3841 a(\(i\)) p 559 3841 a(By) p 706 3841 a(\(i\)) p 835 3841 a(and) p 1019 3841 a(\(ii\)) p 1176 3841 a(of) p 1281 3841 a(Lemma) p 1623 3841 a(6.1,) p 1802 3841 a(w) m(e) p 1940 3841 a(ha) m(v) m(e) p Fn 2159 3841 a(A) p Fm 2232 3856 a(N) p Fg 2299 3799 a(\003) p 2339 3763 141 4 v Fn 2339 3841 a(A) p Fm 2412 3856 a(N) p Fk 2507 3841 a(\033) p Fn 2612 3841 a(A) p Fg 2685 3794 a(y) p Fm 2685 3866 a(N) p Fn 2752 3841 a(A) p Fo 2853 3841 a(=) p Fn 2957 3841 a(L) p Fm 3023 3856 a(N) p Fk 3100 3841 a(\000) p Fn 3187 3841 a(B) p Fo 3293 3841 a(and) p Fn 3477 3841 a(A) p Fg 3550 3794 a(y) p Fm 3550 3866 a(N) p Fg 3617 3774 a(\003) p 3656 3738 V Fn 3656 3841 a(A) p Fg 3729 3794 a(y) p Fm 3729 3866 a(N) p Fk 3825 3841 a(\033) p Fn 0 3984 a(A) p Fm 73 3999 a(N) p Fn 140 3984 a(A) p Fg 213 3937 a(y) p Fm 213 4009 a(N) p Fo 308 3984 a(=) p Fn 412 3984 a(L) p Fm 478 3999 a(N) p Fo 561 3984 a(+) p Fn 652 3984 a(B) p Fo 760 3984 a(in) p 871 3984 a(the) p 1035 3984 a(op) s(erator) p 1425 3984 a(sense.) p 1712 3984 a(By) p 1862 3984 a(Lemma) p 2207 3984 a(6.2,) p Fn 2388 3984 a(A) p Fm 2461 3999 a(N) p Fg 2529 3943 a(\003) p 2568 3906 V Fn 2568 3984 a(A) p Fm 2641 3999 a(N) p Fo 2738 3984 a(and) p Fn 2924 3984 a(A) p Fg 2997 3937 a(y) p Fm 2997 4009 a(N) p Fg 3064 3917 a(\003) p 3104 3881 V Fn 3104 3984 a(A) p Fg 3177 3937 a(y) p Fm 3177 4009 a(N) p Fo 3273 3984 a(are) p 3433 3984 a(self-adjoin) m(t) 0 4105 y(op) s(erators.) p 467 4105 a(Th) m(us,) p 737 4105 a(there) p 980 4105 a(are) p 1137 4105 a(self-adjoin) m(t) p 1633 4105 a(extensions) p Fn 2097 4105 a(X) p Fo 2212 4105 a(and) p Fn 2396 4105 a(Y) p Fo 2500 4105 a(of) p Fn 2606 4105 a(L) p Fm 2672 4120 a(N) p Fo 2766 4105 a(suc) m(h) p 2980 4105 a(that) p Fn 3185 4105 a(A) p Fm 3258 4120 a(N) p Fg 3326 4063 a(\003) p 3365 4027 V Fn 3365 4105 a(A) p Fm 3438 4120 a(N) p Fo 3533 4105 a(=) p Fn 3637 4105 a(X) p Fk 3735 4105 a(\000) p Fn 3822 4105 a(B) p Fo 0 4242 a(and) p Fn 190 4242 a(A) p Fg 263 4195 a(y) p Fm 263 4267 a(N) p Fg 330 4175 a(\003) p 369 4139 V Fn 369 4242 a(A) p Fg 442 4195 a(y) p Fm 442 4267 a(N) p Fo 538 4242 a(=) p Fn 641 4242 a(Y) p Fo 741 4242 a(+) p Fn 839 4242 a(B) p Fo 918 4242 a(.) 146 4363 y(W) p 238 4363 a(e) p 325 4363 a(shall) p 565 4363 a(sho) m(w) p Fn 817 4363 a(X) p Fo 953 4363 a(=) p Fn 1075 4363 a(H) p Fm 1164 4327 a(AB) 1156 4387 y(N) p Fo 1277 4363 a(.) p 1381 4363 a(By) p 1545 4363 a(\(5.29\)) p 1838 4363 a(and) p 2039 4363 a(\(5.30\),) p 2361 4363 a(w) m(e) p 2516 4363 a(see) p 2685 4363 a(that) p 2907 4363 a(the) p 3086 4363 a(graph) p 3374 4363 a(norm) p 3639 4363 a(of) p Fn 3762 4363 a(A) p Fm 3835 4378 a(N) p Fo 0 4483 a(is) p 111 4483 a(equiv) p 333 4483 a(alen) m(t) p 587 4483 a(to) p 719 4483 a(the) p 900 4483 a(norm) p 1167 4483 a(of) p 1291 4483 a(the) p 1472 4483 a(form) p 1715 4483 a(domain) p Fn 2074 4483 a(Q) p Fo(\() p Fn(H) p Fm 2278 4447 a(AB) 2270 4508 y(N) p Fo 2391 4483 a(\).) p 2538 4483 a(Moreo) m(v) m(er,) p Fn 3009 4483 a(D) p Fo 3093 4483 a(\() p Fn(A) p Fm 3204 4498 a(N) p Fo 3271 4483 a(\)) p 3358 4483 a(=) p Fn 3484 4483 a(D) p Fo 3568 4483 a(\() p Fn(L) p Fm 3672 4498 a(N) p Fo 3739 4483 a(\)) p 3826 4483 a(=) p Fn 0 4604 a(C) p Fg 77 4567 a(1) p Fl 70 4628 a(0) p Fo 152 4604 a(\() p Fj(R) p Fl 277 4562 a(2) p Fk 348 4604 a(n) p Fn 429 4604 a(S) p Fm 489 4619 a(N) p Fo 557 4604 a(\)) p 641 4604 a(is) p 753 4604 a(a) p 848 4604 a(form) p 1092 4604 a(core) p 1312 4604 a(for) p Fn 1475 4604 a(D) p Fo 1559 4604 a(\() p Fn(H) p Fm 1686 4567 a(AB) 1678 4628 y(N) p Fo 1798 4604 a(\),) p 1913 4604 a(since) p Fn 2166 4604 a(H) p Fm 2255 4567 a(AB) 2247 4628 y(N) p Fo 2415 4604 a(is) p 2526 4604 a(the) p 2708 4604 a(F) p 2764 4604 a(riedric) m(hs) p 3171 4604 a(extension) p 3616 4604 a(of) p Fn 3741 4604 a(L) p Fm 3807 4619 a(N) p Fo 3875 4604 a(.) 0 4724 y(T) p 62 4724 a(aking) p 328 4724 a(closure,) p 683 4724 a(w) m(e) p 829 4724 a(see) p 990 4724 a(that) p Fn 1203 4724 a(D) p Fo 1287 4724 a(\() p 1325 4646 V Fn(A) p Fm 1398 4739 a(N) p Fo 1465 4724 a(\)) p 1535 4724 a(=) p Fn 1643 4724 a(Q) p Fo(\() p Fn(H) p Fm 1847 4688 a(AB) 1839 4749 y(N) p Fo 1960 4724 a(\).) p 2076 4724 a(According) p 2541 4724 a(to) p 2663 4724 a(the) p 2833 4724 a(pro) s(of) p 3090 4724 a(of) p 3204 4724 a([Re-Si,) p 3523 4724 a(Theorem) 0 4844 y(X.25],) p Fn 283 4844 a(D) p Fo 367 4844 a(\() p 405 4766 V Fn(A) p Fm 478 4859 a(N) p Fo 545 4844 a(\)) p 613 4844 a(equals) p 907 4844 a(to) p 1024 4844 a(the) p 1190 4844 a(form) p 1418 4844 a(domain) p 1763 4844 a(of) p Fn 1872 4844 a(A) p Fm 1945 4859 a(N) p Fg 2012 4803 a(\003) p 2052 4766 V Fn 2052 4844 a(A) p Fm 2125 4859 a(N) p Fo 2192 4844 a(,) p 2250 4844 a(whic) m(h) p 2527 4844 a(equals) p 2821 4844 a(to) p 2938 4844 a(the) p 3104 4844 a(form) p 3332 4844 a(domain) p 3677 4844 a(of) p Fn 3786 4844 a(X) p Fo 3875 4844 a(.) 0 4965 y(Th) m(us) p Fn 247 4965 a(H) p Fm 336 4929 a(AB) 328 4989 y(N) p Fo 482 4965 a(and) p Fn 671 4965 a(X) p Fo 793 4965 a(ha) m(v) m(e) p 1017 4965 a(the) p 1185 4965 a(same) p 1430 4965 a(form) p 1660 4965 a(domain) p 2007 4965 a(with) p 2229 4965 a(the) p 2397 4965 a(form) p 2627 4965 a(core) p Fn 2833 4965 a(C) p Fg 2910 4929 a(1) p Fl 2903 4989 a(0) p Fo 2985 4965 a(\() p Fj(R) p Fl 3110 4923 a(2) p Fk 3172 4965 a(n) p Fn 3244 4965 a(S) p Fm 3304 4980 a(N) p Fo 3371 4965 a(\).) p 3479 4965 a(Moreo) m(v) m(er,) 1902 5214 y(40) p eop %%Page: 41 41 41 40 bop Fo 146 631 a(w) m(e) p 280 631 a(ha) m(v) m(e) p 494 631 a(\() p Fn(X) p 621 631 a(u;) p 721 631 a(u) p Fo(\)) p 841 631 a(=) p 945 631 a(\() p Fk(L) p Fg 1052 646 a(N) p Fn 1123 631 a(u;) p 1223 631 a(u) p Fo(\)) p 1344 631 a(=) p 1447 631 a(\() p Fn(H) p Fm 1574 595 a(AB) 1566 656 y(N) p Fn 1687 631 a(u;) p 1787 631 a(u) p Fo(\)) p 1902 631 a(for) p 2041 631 a(an) m(y) p 2215 631 a(function) p Fn 2587 631 a(u) p Fo 2664 631 a(in) p 2768 631 a(the) p 2926 631 a(form) p 3146 631 a(core) p Fn 3341 631 a(C) p Fg 3418 595 a(1) p Fl 3411 656 a(0) p Fo 3493 631 a(\() p Fj(R) p Fl 3618 589 a(2) p Fk 3659 631 a(n) p Fn 3710 631 a(S) p Fm 3770 646 a(N) p Fo 3837 631 a(\).) 0 751 y(Th) m(us) p Fn 247 751 a(X) p Fo 368 751 a(and) p Fn 558 751 a(H) p Fm 647 715 a(AB) 639 776 y(N) p Fo 793 751 a(ha) m(v) m(e) p 1018 751 a(the) p 1186 751 a(same) p 1430 751 a(quadratic) p 1866 751 a(form.) p 2135 751 a(This) p 2357 751 a(implies) p Fn 2688 751 a(X) p Fo 2805 751 a(=) p Fn 2908 751 a(H) p Fm 2997 715 a(AB) 2989 776 y(N) p Fo 3110 751 a(.) p 3181 751 a(Moreo) m(v) m(er,) p Fn 908 944 a(D) p Fo 992 944 a(\() p 1030 866 141 4 v Fn(A) p Fm 1103 959 a(N) p Fo 1170 944 a(\)) p Fk 1230 944 a(\\) p Fn 1318 944 a(D) p Fo 1402 944 a(\() p Fn(L) p Fg 1506 903 a(\003) p Fm 1506 969 a(N) p Fo 1573 944 a(\)) p 1639 944 a(=) p Fn 1743 944 a(Q) p Fo(\() p Fn(H) p Fm 1947 903 a(AB) 1939 969 y(N) p Fo 2060 944 a(\)) p Fk 2120 944 a(\\) p Fn 2208 944 a(D) p Fo 2292 944 a(\() p Fn(L) p Fg 2396 903 a(\003) p Fm 2396 969 a(N) p Fo 2464 944 a(\)) p 2529 944 a(=) p Fn 2633 944 a(D) p Fo 2717 944 a(\() p Fn(H) p Fm 2844 903 a(AB) 2836 969 y(N) p Fo 2957 944 a(\)) 0 1137 y(since) p Fn 240 1137 a(H) p Fm 329 1101 a(AB) 321 1161 y(N) p Fo 475 1137 a(is) p 574 1137 a(the) p 743 1137 a(F) p 799 1137 a(riedric) m(hs) p 1192 1137 a(extension) p 1625 1137 a(of) p Fn 1736 1137 a(L) p Fm 1802 1152 a(N) p Fo 1870 1137 a(.) p 1943 1137 a(Th) m(us) p 2191 1137 a(\(6.1\)) p 2424 1137 a(follo) m(ws) p 2745 1137 a(from) p 2977 1137 a(\(i\)) p 3113 1137 a(of) p 3225 1137 a(Prop) s(osition) p 3750 1137 a(5.7.) 0 1280 y(The) p 201 1280 a(pro) s(of) p 455 1280 a(of) p Fn 566 1280 a(Y) p Fo 672 1280 a(=) p Fn 776 1280 a(H) p Fm 865 1244 a(AB) 857 1305 y(N) p Fo 1010 1280 a(and) p Fn 1200 1280 a(D) p Fo 1284 1280 a(\() p 1322 1177 V Fn(A) p Fg 1395 1233 a(y) p Fm 1395 1304 a(N) p Fo 1462 1280 a(\)) p Fk 1522 1280 a(\\) p Fn 1611 1280 a(D) p Fo 1695 1280 a(\() p Fn(L) p Fg 1799 1244 a(\003) p Fm 1799 1305 a(N) p Fo 1866 1280 a(\)) p 1932 1280 a(=) p Fn 2035 1280 a(D) p Fo 2119 1280 a(\() p Fn(H) p Fm 2246 1244 a(AB) 2238 1305 y(N) p Fo 2359 1280 a(\)) p 2429 1280 a(is) p 2527 1280 a(similar.) 146 1400 y(\(ii\)) p 314 1400 a(One) p 526 1400 a(can) p 710 1400 a(pro) m(v) m(e) p 978 1400 a(this) p 1174 1400 a(assertion) p 1587 1400 a(b) m(y) p 1727 1400 a(using) p 1988 1400 a(the) p 2162 1400 a(argumen) m(t) p 2603 1400 a(similar) p 2929 1400 a(to) p 3053 1400 a(the) p 3227 1400 a(\014rst) p 3433 1400 a(part) p 3650 1400 a(of) p 3767 1400 a(the) 0 1521 y(pro) s(of) p 255 1521 a(of) p 366 1521 a(\(i\).) 146 1641 y(\(iii\)) p 336 1641 a(By) p 489 1641 a(Prop) s(osition) p 1014 1641 a(5.4,) p 1198 1641 a(the) p 1366 1641 a(op) s(erator) p 1759 1641 a(domain) p 2106 1641 a(of) p Fn 2217 1641 a(H) p Fl 2306 1600 a(+) p Fm 2298 1665 a(N) p Fo 2398 1641 a(and) p Fn 2587 1641 a(H) p Fg 2676 1600 a(\000) p Fm 2668 1665 a(N) p Fo 2768 1641 a(is) p 2866 1641 a(decomp) s(osed) p 3411 1641 a(as) p Fn 1004 1846 a(D) p Fo 1088 1846 a(\() p Fn(H) p Fl 1215 1805 a(+) p Fm 1207 1871 a(N) p Fo 1273 1846 a(\)) p 1394 1846 a(=) p Fn 1553 1846 a(D) p Fo 1637 1846 a(\() p 1675 1768 134 4 v Fn(L) p Fm 1741 1861 a(N) p Fo 1809 1846 a(\)) p Fk 1869 1846 a(\010) p Fm 1946 1861 a(al) q(g) p Fi 2068 1750 a(\020) p Fo 2117 1846 a(Ran) p Fn 2309 1846 a(E) p Fk 2409 1846 a(\\) p Fn 2497 1846 a(D) p Fo 2581 1846 a(\() p Fn(H) p Fl 2708 1805 a(+) p Fm 2700 1871 a(N) p Fo 2767 1846 a(\)) p Fi 2805 1750 a(\021) p Fn 1004 2029 a(D) p Fo 1088 2029 a(\() p Fn(H) p Fg 1215 1988 a(\000) p Fm 1207 2054 a(N) p Fo 1273 2029 a(\)) p 1394 2029 a(=) p Fn 1553 2029 a(D) p Fo 1637 2029 a(\() p 1675 1951 V Fn(L) p Fm 1741 2044 a(N) p Fo 1809 2029 a(\)) p Fk 1869 2029 a(\010) p Fm 1946 2044 a(al) q(g) p Fi 2068 1933 a(\020) p Fo 2117 2029 a(Ran) p Fn 2309 2029 a(E) p Fk 2409 2029 a(\\) p Fn 2497 2029 a(D) p Fo 2581 2029 a(\() p Fn(H) p Fg 2708 1988 a(\000) p Fm 2700 2054 a(N) p Fo 2767 2029 a(\)) p Fi 2805 1933 a(\021) p Fn 2871 2029 a(:) p Fo 0 2234 a(T) p 62 2234 a(ak) m(e) p Fn 242 2234 a(u) p Fk 335 2234 a(2) p Fo 439 2234 a(Ran) p Fn 630 2234 a(E) p Fo 708 2234 a(.) p Fn 796 2234 a(u) p Fo 890 2234 a(is) p 994 2234 a(written) p 1341 2234 a(as) p 1467 2234 a(\(5.34\).) p 1804 2234 a(Then) p Fn 2065 2234 a(u) p Fo 2158 2234 a(satis\014es) p 2524 2234 a(the) p 2697 2234 a(conditions) p Fn 3170 2234 a(u) p Fk 3263 2234 a(2) p Fn 3367 2234 a(D) p Fo 3451 2234 a(\() p Fn(A) p Fm 3562 2249 a(N) p Fg 3629 2192 a(\003) p Fo 3669 2234 a(\)) p 3745 2234 a(and) p Fn 0 2355 a(A) p Fm 73 2370 a(N) p Fg 140 2313 a(\003) p Fn 180 2355 a(u) p Fo 263 2355 a(=) p Fk 367 2355 a(A) p Fg 447 2307 a(y) p Fm 447 2379 a(N) p Fn 513 2355 a(u) p Fk 597 2355 a(2) p Fn 691 2355 a(D) p Fo 775 2355 a(\() p 813 2276 141 4 v Fn(A) p Fm 886 2370 a(N) p Fo 953 2355 a(\)) p 1023 2355 a(if) p 1112 2355 a(and) p 1302 2355 a(only) p 1516 2355 a(if) p Fn 1139 2547 a(c) p Fl 1181 2562 a(4) p Fm(j) p Fg 1249 2562 a(\000) p Fl(2) p Fo 1371 2547 a(=) p Fn 1474 2547 a(c) p Fl 1516 2562 a(4) p Fm(j) p Fo 1616 2547 a(=) p 1719 2547 a(0) p 1800 2547 a(for) p 1950 2547 a(ev) m(ery) p Fn 2208 2547 a(j) p Fo 2282 2547 a(=) p 2385 2547 a(1) p Fn(;) p 2478 2547 a(:) p 2522 2547 a(:) p 2566 2547 a(:) p 2609 2547 a(;) p 2653 2547 a(N) p 2736 2547 a(;) p Fo 0 2740 a(b) m(y) p 145 2740 a(\(ii\)) p 318 2740 a(of) p 439 2740 a(Lemma) p 797 2740 a(6.1,) p 993 2740 a(\(6.1\),) p 1266 2740 a(\(5.36\)) p 1558 2740 a(and) p 1757 2740 a(the) p 1935 2740 a(argumen) m(t) p 2381 2740 a(follo) m(wing) p 2803 2740 a(\(5.36\).) p 3152 2740 a(Th) m(us) p 3409 2740 a(\(6.2\)) p 3652 2740 a(holds.) 0 2883 y(Similarly) p 378 2883 a(,) p Fn 434 2883 a(u) p Fo 522 2883 a(satis\014es) p 881 2883 a(the) p 1049 2883 a(conditions) p Fn 1516 2883 a(u) p Fk 1599 2883 a(2) p Fn 1693 2883 a(D) p Fo 1777 2883 a(\() p Fn(A) p Fg 1888 2836 a(y) p Fm 1888 2908 a(N) p Fg 1955 2816 a(\003) p Fo 1994 2883 a(\)) p 2065 2883 a(and) p Fn 2254 2883 a(A) p Fg 2327 2836 a(y) p Fm 2327 2908 a(N) p Fg 2395 2816 a(\003) p Fn 2434 2883 a(u) p Fo 2517 2883 a(=) p Fk 2621 2883 a(A) p Fm 2701 2898 a(N) p Fn 2768 2883 a(u) p Fk 2851 2883 a(2) p Fn 2945 2883 a(D) p Fo 3029 2883 a(\() p 3067 2780 V Fn(A) p Fg 3140 2836 a(y) p Fm 3140 2908 a(N) p Fo 3207 2883 a(\)) p 3277 2883 a(if) p 3367 2883 a(and) p 3557 2883 a(only) p 3771 2883 a(if) p Fn 1094 3076 a(c) p Fl 1136 3091 a(4) p Fm(j) p Fg 1204 3091 a(\000) p Fl(3) p Fo 1326 3076 a(=) p Fn 1429 3076 a(c) p Fl 1471 3091 a(4) p Fm(j) p Fg 1539 3091 a(\000) p Fl(1) p Fo 1661 3076 a(=) p 1764 3076 a(0) p 1846 3076 a(for) p 1995 3076 a(ev) m(ery) p Fn 2253 3076 a(j) p Fo 2327 3076 a(=) p 2431 3076 a(1) p Fn(;) p 2524 3076 a(:) p 2568 3076 a(:) p 2612 3076 a(:) p 2654 3076 a(;) p 2698 3076 a(N) p 2781 3076 a(;) p Fo 0 3269 a(b) m(y) p 135 3269 a(\(ii\)) p 298 3269 a(of) p 409 3269 a(Lemma) p 757 3269 a(6.1,) p 942 3269 a(\(6.1\),) p 1202 3269 a(\(5.35\)) p 1484 3269 a(and) p 1673 3269 a(the) p 1841 3269 a(argumen) m(t) p 2278 3269 a(follo) m(wing) p 2689 3269 a(\(5.35\).) p 3009 3269 a(Th) m(us) p 3256 3269 a(\(6.3\)) p 3489 3269 a(holds.) 146 3389 y(\(iv\)) p 333 3389 a(This) p 556 3389 a(assertion) p 964 3389 a(follo) m(ws) p 1284 3389 a(from) p 1514 3389 a(\(i\)) p 1650 3389 a(of) p 1761 3389 a(Prop) s(osition) p 2286 3389 a(5.7) p 2443 3389 a(and) p 2632 3389 a(\(iii\)) p 2822 3389 a(of) p 2933 3389 a(this) p 3123 3389 a(prop) s(osition.) p 3674 3321 64 4 v 3674 3414 4 93 v 3734 3414 V 3674 3417 64 4 v 0 3586 a(Remark.) p 448 3586 a(With) p 715 3586 a(the) p 897 3586 a(notation) p 1302 3586 a(of) p 1427 3586 a(Prop) s(osition) p 1966 3586 a(5.4,) p Fn 2168 3586 a(H) p Fl 2257 3545 a(+) p Fm 2249 3610 a(N) p Fo 2317 3586 a(,) p Fn 2394 3586 a(H) p Fg 2483 3545 a(\000) p Fm 2475 3610 a(N) p Fo 2589 3586 a(are) p 2766 3586 a(the) p 2948 3586 a(self-adjoin) m(t) p 3465 3586 a(extensions) 0 3707 y(corresp) s(onding) p 626 3707 a(to) p 746 3707 a(the) p 914 3707 a(orthogonal) p 1401 3707 a(pro) p 1547 3707 a(jection) p 1865 3707 a(matrices) p Fn 2256 3707 a(M) p Fl 2360 3665 a(+) p Fm 2350 3731 a(N) p Fo 2419 3707 a(,) p Fn 2479 3707 a(M) p Fg 2583 3665 a(\000) p Fm 2573 3731 a(N) p Fo 2643 3707 a(,) p 2702 3707 a(where) p Fn 622 4077 a(M) p Fl 726 4036 a(+) p Fm 716 4102 a(N) p Fo 869 4077 a(=) p Fi 1028 3831 a(0) 1028 3978 y(B) 1028 4027 y(B) 1028 4077 y(B) 1028 4130 y(@) p Fn 1142 3896 a(M) p Fl 1246 3860 a(+) p Fn 1432 3896 a(O) p 1635 3896 a(:) p 1679 3896 a(:) p 1723 3896 a(:) p 1876 3896 a(O) 1185 4016 y(O) p 1389 4016 a(M) p Fl 1493 3980 a(+) p Fn 1635 4016 a(:) p 1679 4016 a(:) p 1723 4016 a(:) p 1876 4016 a(O) p Fo 1170 4137 a(.) p 1212 4137 a(.) p 1255 4137 a(.) p 1298 4137 a(.) p 1341 4137 a(.) p 1384 4137 a(.) p 1427 4137 a(.) p 1470 4137 a(.) p 1513 4137 a(.) p 1556 4137 a(.) p 1599 4137 a(.) p 1642 4137 a(.) p 1685 4137 a(.) p 1728 4137 a(.) p 1770 4137 a(.) p 1813 4137 a(.) p 1856 4137 a(.) p 1899 4137 a(.) p 1942 4137 a(.) p Fn 1185 4257 a(O) p 1432 4257 a(O) p 1635 4257 a(:) p 1679 4257 a(:) p 1723 4257 a(:) p 1833 4257 a(M) p Fl 1937 4221 a(+) p Fi 2038 3831 a(1) 2038 3978 y(C) 2038 4027 y(C) 2038 4077 y(C) 2038 4130 y(A) p Fn 2128 4077 a(;) p 2269 4077 a(M) p Fl 2373 4036 a(+) p Fo 2460 4077 a(=) p Fi 2564 3831 a(0) 2564 3978 y(B) 2564 4027 y(B) 2564 4077 y(B) 2564 4130 y(@) p Fo 2678 3896 a(0) p 2810 3896 a(0) p 2942 3896 a(0) p 3073 3896 a(0) 2678 4016 y(0) p 2810 4016 a(1) p 2942 4016 a(0) p 3073 4016 a(0) 2678 4137 y(0) p 2810 4137 a(0) p 2942 4137 a(0) p 3073 4137 a(0) 2678 4257 y(0) p 2810 4257 a(0) p 2942 4257 a(0) p 3073 4257 a(1) p Fi 3164 3831 a(1) 3164 3978 y(C) 3164 4027 y(C) 3164 4077 y(C) 3164 4130 y(A) p Fn 622 4592 a(M) p Fg 726 4551 a(\000) p Fm 716 4617 a(N) p Fo 869 4592 a(=) p Fi 1028 4346 a(0) 1028 4492 y(B) 1028 4542 y(B) 1028 4592 y(B) 1028 4645 y(@) p Fn 1142 4411 a(M) p Fg 1246 4374 a(\000) p Fn 1432 4411 a(O) p 1635 4411 a(:) p 1679 4411 a(:) p 1723 4411 a(:) p 1876 4411 a(O) 1185 4531 y(O) p 1389 4531 a(M) p Fg 1493 4495 a(\000) p Fn 1635 4531 a(:) p 1679 4531 a(:) p 1723 4531 a(:) p 1876 4531 a(O) p Fo 1170 4651 a(.) p 1212 4651 a(.) p 1255 4651 a(.) p 1298 4651 a(.) p 1341 4651 a(.) p 1384 4651 a(.) p 1427 4651 a(.) p 1470 4651 a(.) p 1513 4651 a(.) p 1556 4651 a(.) p 1599 4651 a(.) p 1642 4651 a(.) p 1685 4651 a(.) p 1728 4651 a(.) p 1770 4651 a(.) p 1813 4651 a(.) p 1856 4651 a(.) p 1899 4651 a(.) p 1942 4651 a(.) p Fn 1185 4772 a(O) p 1432 4772 a(O) p 1635 4772 a(:) p 1679 4772 a(:) p 1723 4772 a(:) p 1833 4772 a(M) p Fg 1937 4736 a(\000) p Fi 2038 4346 a(1) 2038 4492 y(C) 2038 4542 y(C) 2038 4592 y(C) 2038 4645 y(A) p Fn 2128 4592 a(;) p 2269 4592 a(M) p Fg 2373 4551 a(\000) p Fo 2460 4592 a(=) p Fi 2564 4346 a(0) 2564 4492 y(B) 2564 4542 y(B) 2564 4592 y(B) 2564 4645 y(@) p Fo 2678 4411 a(1) p 2810 4411 a(0) p 2942 4411 a(0) p 3073 4411 a(0) 2678 4531 y(0) p 2810 4531 a(0) p 2942 4531 a(0) p 3073 4531 a(0) 2678 4651 y(0) p 2810 4651 a(0) p 2942 4651 a(1) p 3073 4651 a(0) 2678 4772 y(0) p 2810 4772 a(0) p 2942 4772 a(0) p 3073 4772 a(0) p Fi 3164 4346 a(1) 3164 4492 y(C) 3164 4542 y(C) 3164 4592 y(C) 3164 4645 y(A) p Fn 3253 4592 a(:) p Fo 0 4965 a(The) p 201 4965 a(matrices) p Fn 591 4965 a(M) p Fg 695 4929 a(\006) p Fo 788 4965 a(satisfy) p 1092 4965 a(\(5.24\).) 1902 5214 y(41) p eop %%Page: 42 42 42 41 bop Fh 0 631 a(7) p 242 631 a(Pro) t(of) p 687 631 a(of) p 871 631 a(Theorem) p 1555 631 a(1.2) p Fo 0 850 a(In) p 122 850 a(this) p 312 850 a(section,) p 665 850 a(w) m(e) p 809 850 a(shall) p 1037 850 a(pro) m(v) m(e) p 1300 850 a(Theorem) p 1712 850 a(1.2.) p 1908 850 a(The) p 2108 850 a(second) p 2423 850 a(assertion) p 2831 850 a(of) p 2942 850 a(the) p 3110 850 a(follo) m(wing) p 3522 850 a(lemma) p 3836 850 a(is) 0 970 y(an) p 135 970 a(immediate) p 612 970 a(corollary) p 1016 970 a(of) p 1127 970 a([W) p 1246 970 a(e,) p 1349 970 a(Exercise) p 1733 970 a(8.8].) p 1956 970 a(W) p 2048 970 a(e) p 2124 970 a(shall) p 2352 970 a(giv) m(e) p 2552 970 a(a) p 2633 970 a(pro) s(of) p 2888 970 a(for) p 3037 970 a(the) p 3205 970 a(completeness.) p Fq 146 1174 a(Lemma) p 543 1174 a(7.1) p Ff 735 1174 a(L) p 791 1174 a(et) p Fn 913 1174 a(L) p Ff 1023 1174 a(b) p 1063 1174 a(e) p 1152 1174 a(a) p 1246 1174 a(symmetric) p 1730 1174 a(op) p 1825 1174 a(er) p 1906 1174 a(ators) p 2163 1174 a(on) p 2312 1174 a(a) p 2405 1174 a(Hilb) p 2572 1174 a(ert) p 2735 1174 a(sp) p 2820 1174 a(ac) p 2910 1174 a(e) p Fk 2998 1174 a(H) p Ff 3083 1174 a(.) p 3186 1174 a(Supp) p 3388 1174 a(ose) p 3566 1174 a(that) p 3775 1174 a(the) 0 1294 y(de\014ciency) p 446 1294 a(indic) p 651 1294 a(es) p 766 1294 a(of) p Fn 876 1294 a(L) p Ff 973 1294 a(ar) p 1059 1294 a(e) p Fo 1134 1294 a(\() p Fn(n;) p 1274 1294 a(n) p Fo(\)) p Ff 1401 1294 a(and) p Fn 1585 1294 a(n) p Ff 1674 1294 a(is) p 1774 1294 a(\014nite.) p 2064 1294 a(L) p 2120 1294 a(et) p Fn 2228 1294 a(A) p Ff 2332 1294 a(and) p Fn 2517 1294 a(B) p Ff 2626 1294 a(ar) p 2712 1294 a(e) p 2787 1294 a(two) p 2965 1294 a(self-adjoint) p 3466 1294 a(extensions) 0 1415 y(of) p Fn 115 1415 a(L) p Ff(.) p 256 1415 a(Then,) p 540 1415 a(the) p 701 1415 a(fol) p 811 1415 a(lowing) p 1115 1415 a(holds.) 146 1535 y(\(i\)) p 291 1535 a(We) p 468 1535 a(have) p Fn 692 1535 a(\033) p Fm 747 1550 a(ess) p Fo 850 1535 a(\() p Fn(A) p Fo(\)) p 1026 1535 a(=) p Fn 1130 1535 a(\033) p Fm 1185 1550 a(ess) p Fo 1287 1535 a(\() p Fn(B) p Fo 1404 1535 a(\)) p Ff(.) 146 1655 y(\(ii\)) p 328 1655 a(F) p 385 1655 a(or) p 517 1655 a(any) p 711 1655 a(interval) p Fn 1075 1655 a(I) p Fo 1167 1655 a(=) p 1284 1655 a(\() p Fn(c) p Fl 1364 1670 a(1) p Fn 1403 1655 a(;) p 1447 1655 a(c) p Fl 1489 1670 a(2) p Fo 1528 1655 a(\)) p Ff 1608 1655 a(satisfying) p Fn 2049 1655 a(c) p Fl 2091 1670 a(1) p Fn 2171 1655 a(<) p 2288 1655 a(c) p Fl 2330 1670 a(2) p Ff 2411 1655 a(and) p Fo 2608 1655 a(dim) p 2787 1655 a(Ran) p Fn 2978 1655 a(P) p Fm 3041 1670 a(I) p Fo 3081 1655 a(\() p Fn(A) p Fo(\)) p Fn 3271 1655 a(<) p Fk 3388 1655 a(1) p Ff(,) p 3561 1655 a(we) p 3713 1655 a(have) p Fo 0 1776 a(dim) p 179 1776 a(Ran) p Fn 371 1776 a(P) p Fm 434 1791 a(I) p Fo 473 1776 a(\() p Fn(B) p Fo 590 1776 a(\)) p Fn 656 1776 a(<) p Fk 760 1776 a(1) p Ff 894 1776 a(and) p Fk 1123 1996 a(j) p Fo 1168 1996 a(dim) p 1347 1996 a(Ran) p Fn 1538 1996 a(P) p Fm 1601 2011 a(I) p Fo 1641 1996 a(\() p Fn(A) p Fo(\)) p Fk 1812 1996 a(\000) p Fo 1912 1996 a(dim) p 2091 1996 a(Ran) p Fn 2282 1996 a(P) p Fm 2345 2011 a(I) p Fo 2385 1996 a(\() p Fn(B) p Fo 2502 1996 a(\)) p Fk(j) p 2596 1996 a(\024) p Fn 2701 1996 a(d;) p Ff 0 2216 a(wher) p 196 2216 a(e) p Fn 634 2336 a(d) p Fo 713 2336 a(=) p 816 2336 a(dim) p Fn 995 2336 a(D) p Fo 1079 2336 a(\() p Fn(A) p Fo(\)) p Fn(=) p Fo 1294 2336 a(\() p Fn 1331 2336 a(D) p Fo 1415 2336 a(\() p Fn(A) p Fo(\)) p Fk 1586 2336 a(\\) p Fn 1675 2336 a(D) p Fo 1759 2336 a(\() p Fn(B) p Fo 1876 2336 a(\)\)) p 1979 2336 a(=) p 2083 2336 a(dim) p Fn 2262 2336 a(D) p Fo 2346 2336 a(\() p Fn(B) p Fo 2463 2336 a(\)) p Fn(=) p Fo 2567 2336 a(\() p Fn 2604 2336 a(D) p Fo 2688 2336 a(\() p Fn(A) p Fo(\)) p Fk 2859 2336 a(\\) p Fn 2947 2336 a(D) p Fo 3031 2336 a(\() p Fn(B) p Fo 3148 2336 a(\)\)) p Fn 3241 2336 a(:) p Fo 0 2540 a(Remark:) p 405 2540 a(The) p 606 2540 a(inequalit) m(y) p Fn 1058 2540 a(d) p Fk 1136 2540 a(\024) p Fn 1242 2540 a(n) p Fo 1332 2540 a(alw) m(a) m(ys) p 1645 2540 a(holds,) p 1927 2540 a(since) p Fn 2166 2540 a(A) p Fo 2272 2540 a(and) p Fn 2461 2540 a(B) p Fo 2573 2540 a(are) p Fn 2735 2540 a(n) p Fo 2826 2540 a(rank) p 3051 2540 a(extension) p 3483 2540 a(of) p 3594 2461 67 4 v Fn 3594 2540 a(L) p Fo(,) p Ff 146 2768 a(Pr) p 248 2768 a(o) p 293 2768 a(of.) p Fo 435 2768 a(\(i\)) p 571 2768 a(See) p 744 2768 a([W) p 863 2768 a(e,) p 967 2768 a(Theorem) p 1378 2768 a(8.17].) 146 2888 y(\(ii\)) p 309 2888 a(First) p 547 2888 a(w) m(e) p 690 2888 a(sho) m(w) 1160 3009 y(dim) p 1339 3009 a(Ran) p Fn 1531 3009 a(P) p Fm 1594 3024 a(I) p Fo 1634 3009 a(\() p Fn(B) p Fo 1751 3009 a(\)) p Fk 1816 3009 a(\024) p Fo 1921 3009 a(dim) p 2101 3009 a(Ran) p Fn 2292 3009 a(P) p Fm 2355 3024 a(I) p Fo 2395 3009 a(\() p Fn(A) p Fo(\)) p 2566 3009 a(+) p Fn 2664 3009 a(d:) p Fo 3701 3009 a(\(7.1\)) 0 3183 y(Supp) s(ose) 1175 3303 y(dim) p 1354 3303 a(Ran) p Fn 1545 3303 a(P) p Fm 1608 3318 a(I) p Fo 1648 3303 a(\() p Fn(B) p Fo 1765 3303 a(\)) p Fn 1831 3303 a(>) p Fo 1934 3303 a(dim) p 2113 3303 a(Ran) p Fn 2305 3303 a(P) p Fm 2368 3318 a(I) p Fo 2408 3303 a(\() p Fn(A) p Fo(\)) p 2579 3303 a(+) p Fn 2677 3303 a(d) p Fo 0 3478 a(holds.) p 293 3478 a(Put) p Fn 484 3478 a(X) p Fo 601 3478 a(=) p 704 3478 a(Ran) p Fn 896 3478 a(P) p Fm 959 3493 a(I) p Fo 998 3478 a(\() p Fn(B) p Fo 1115 3478 a(\)) p Fk 1176 3478 a(\\) p Fo 1264 3478 a(\(Ran) p Fn 1493 3478 a(P) p Fm 1556 3493 a(I) p Fo 1596 3478 a(\() p Fn(A) p Fo(\)\)) p Fg 1783 3441 a(?) p Fo 1842 3478 a(.) p 1913 3478 a(Then) p Fn 2167 3478 a(X) p Fk 2284 3478 a(\032) p Fn 2389 3478 a(D) p Fo 2473 3478 a(\() p Fn(B) p Fo 2590 3478 a(\)) p 2660 3478 a(and) p 2850 3478 a(dim) p Fn 3029 3478 a(X) p 3146 3478 a(>) p 3249 3478 a(d) p Fo(.) 146 3598 y(W) p 238 3598 a(e) p 325 3598 a(shall) p 564 3598 a(sho) m(w) p 816 3598 a(that) p Fn 1038 3598 a(X) p Fk 1156 3598 a(\\) p Fn 1252 3598 a(D) p Fo 1336 3598 a(\() p Fn(A) p Fo(\)) p 1530 3598 a(=) p Fk 1652 3598 a(f) p Fo(0) p Fk(g) p Fo(.) p 1902 3598 a(Supp) s(ose) p 2296 3598 a(that) p 2518 3598 a(there) p 2777 3598 a(exists) p 3057 3598 a(nonzero) p 3431 3598 a(v) m(ector) p Fn 3734 3598 a(u) p Fk 3836 3598 a(2) p Fn 0 3718 a(X) p Fk 111 3718 a(\\) p Fn 199 3718 a(D) p Fo 283 3718 a(\() p Fn(A) p Fo(\).) p 503 3718 a(Since) p Fn 757 3718 a(u) p Fk 841 3718 a(2) p Fo 935 3718 a(\(Ran) p Fn 1164 3718 a(P) p Fm 1227 3733 a(I) p Fo 1267 3718 a(\() p Fn(A) p Fo(\)\)) p Fg 1454 3682 a(?) p Fk 1535 3718 a(\\) p Fn 1623 3718 a(D) p Fo 1707 3718 a(\() p Fn(A) p Fo(\),) p 1916 3718 a(w) m(e) p 2059 3718 a(ha) m(v) m(e) p Fk 1470 3938 a(jj) p Fo(\() p Fn(A) p Fk 1658 3938 a(\000) p Fn 1758 3938 a(c) p Fo(\)) p Fn(u) p Fk(jj) p Fl 1950 3897 a(2) p Fk 2016 3938 a(\025) p Fn 2121 3938 a(\017) p Fl 2160 3897 a(2) p Fk 2200 3938 a(jj) p Fn(u) p Fk(jj) p Fl 2368 3897 a(2) p Fn 2405 3938 a(;) p Fo 0 4158 a(where) p Fn 293 4158 a(c) p Fo 382 4158 a(=) p Fm 514 4119 a(c) p Fd 545 4128 a(1) p Fl 579 4119 a(+) p Fm(c) p Fd 665 4128 a(2) p 514 4135 186 4 v Fl 589 4193 a(2) p Fo 709 4158 a(,) p Fn 783 4158 a(\017) p Fo 869 4158 a(=) p Fm 1002 4119 a(c) p Fd 1033 4128 a(2) p Fg 1067 4119 a(\000) p Fm(c) p Fd 1153 4128 a(1) p 1002 4135 V Fl 1077 4193 a(2) p Fo 1197 4158 a(.) p 1301 4158 a(Ho) m(w) m(ev) m(er,) p 1735 4158 a(since) p Fn 1985 4158 a(L) p Fk 2098 4158 a(\032) p Fn 2222 4158 a(A) p Fk 2342 4158 a(\032) p Fn 2466 4158 a(L) p Fg 2532 4122 a(\003) p Fo 2572 4158 a(,) p Fn 2646 4158 a(L) p Fk 2759 4158 a(\032) p Fn 2883 4158 a(B) p Fk 3009 4158 a(\032) p Fn 3133 4158 a(L) p Fg 3199 4122 a(\003) p Fo 3282 4158 a(and) p 3483 4158 a(since) p Fn 3733 4158 a(u) p Fk 3836 4158 a(2) p Fo 0 4279 a(Ran) p Fn 191 4279 a(P) p Fm 254 4294 a(I) p Fo 294 4279 a(\() p Fn(B) p Fo 411 4279 a(\)) p Fk 471 4279 a(\\) p Fn 560 4279 a(D) p Fo 644 4279 a(\() p Fn(A) p Fo(\),) p 852 4279 a(w) m(e) p 996 4279 a(ha) m(v) m(e) p Fk 815 4499 a(jj) p Fo(\() p Fn(A) p Fk 1004 4499 a(\000) p Fn 1103 4499 a(c) p Fo(\)) p Fn(u) p Fk(jj) p Fl 1295 4458 a(2) p Fo 1361 4499 a(=) p Fk 1465 4499 a(jj) p Fo(\() p Fn(L) p Fg 1625 4458 a(\003) p Fk 1686 4499 a(\000) p Fn 1785 4499 a(c) p Fo(\)) p Fn(u) p Fk(jj) p Fl 1977 4458 a(2) p Fo 2043 4499 a(=) p Fk 2147 4499 a(jj) p Fo(\() p Fn(B) p Fk 2341 4499 a(\000) p Fn 2441 4499 a(c) p Fo(\)) p Fn(u) p Fk(jj) p Fl 2633 4458 a(2) p Fn 2699 4499 a(<) p 2803 4499 a(\017) p Fl 2842 4458 a(2) p Fk 2881 4499 a(jj) p Fn(u) p Fk(jj) p Fl 3049 4458 a(2) p Fo 0 4719 a(\(one) p 214 4719 a(can) p 390 4719 a(sho) m(w) p 629 4719 a(that) p 837 4719 a(the) p 1002 4719 a(strict) p 1255 4719 a(inequalit) m(y) p 1704 4719 a(holds) p 1957 4719 a(b) m(y) p 2089 4719 a(the) p 2254 4719 a(sp) s(ectral) p 2618 4719 a(decomp) s(osition) p 3258 4719 a(theorem\).) p 3712 4719 a(This) 0 4839 y(con) m(tradicts) p 502 4839 a(the) p 670 4839 a(previous) p 1058 4839 a(inequalit) m(y) p 1471 4839 a(.) 1902 5214 y(42) p eop %%Page: 43 43 43 42 bop Fo 146 631 a(Th) m(us,) p 421 631 a(the) p 589 631 a(pro) p 735 631 a(jection) p 1052 631 a(map) p Fn 1144 830 a(X) p Fk 1260 830 a(3) p Fn 1354 830 a(u) p Fk 1437 830 a(7!) p Fo 1565 830 a([) p Fn(u) p Fo(]) p Fk 1702 830 a(2) p Fn 1796 830 a(D) p Fo 1880 830 a(\() p Fn(B) p Fo 1997 830 a(\)) p Fn(=) p Fo 2101 830 a(\() p Fn 2138 830 a(D) p Fo 2222 830 a(\() p Fn(A) p Fo(\)) p Fk 2393 830 a(\\) p Fn 2482 830 a(D) p Fo 2566 830 a(\() p Fn(B) p Fo 2683 830 a(\)\)) 0 1028 y(is) p 98 1028 a(injectiv) m(e.) p 524 1028 a(Hence) p 813 1028 a(w) m(e) p 957 1028 a(ha) m(v) m(e) 1075 1227 y(dim) p Fn 1255 1227 a(D) p Fo 1339 1227 a(\() p Fn(B) p Fo 1456 1227 a(\)) p Fn(=) p Fo 1560 1227 a(\() p Fn 1597 1227 a(D) p Fo 1681 1227 a(\() p Fn(A) p Fo(\)) p Fk 1852 1227 a(\\) p Fn 1940 1227 a(D) p Fo 2024 1227 a(\() p Fn(B) p Fo 2141 1227 a(\)\)) p Fk 2244 1227 a(\025) p Fo 2350 1227 a(dim) p Fn 2529 1227 a(X) p 2645 1227 a(>) p 2749 1227 a(d:) p Fo 0 1426 a(This) p 223 1426 a(con) m(tradicts) p 725 1426 a(the) p 893 1426 a(assumption.) 146 1546 y(Th) m(us) p 393 1546 a(w) m(e) p 537 1546 a(ha) m(v) m(e) p 762 1546 a(pro) m(v) m(ed) p 1079 1546 a(\(7.1\)) p 1312 1546 a(and) p 1502 1546 a(that) p 1713 1546 a(dim) p 1892 1546 a(Ran) p Fn 2084 1546 a(P) p Fm 2147 1561 a(I) p Fo 2186 1546 a(\() p Fn(B) p Fo 2303 1546 a(\)) p 2374 1546 a(is) p 2472 1546 a(\014nite.) p 2759 1546 a(The) p 2960 1546 a(equalit) m(y) 1174 1745 y(dim) p 1353 1745 a(Ran) p Fn 1544 1745 a(P) p Fm 1607 1760 a(I) p Fo 1647 1745 a(\() p Fn(A) p Fo(\)) p Fk 1824 1745 a(\024) p Fo 1929 1745 a(dim) p 2108 1745 a(Ran) p Fn 2300 1745 a(P) p Fm 2363 1760 a(I) p Fo 2402 1745 a(\() p Fn(B) p Fo 2519 1745 a(\)) p 2579 1745 a(+) p Fn 2678 1745 a(d) p Fo 0 1944 a(is) p 98 1944 a(pro) m(v) m(ed) p 415 1944 a(b) m(y) p 551 1944 a(exc) m(hanging) p Fn 1055 1944 a(A) p Fo 1160 1944 a(for) p Fn 1309 1944 a(B) p Fo 1421 1944 a(in) p 1535 1944 a(the) p 1703 1944 a(ab) s(o) m(v) m(e) p 1979 1944 a(pro) s(of.) p 2272 1876 64 4 v 2272 1969 4 93 v 2332 1969 V 2272 1972 64 4 v Ff 146 2148 a(Pr) p 248 2148 a(o) p 293 2148 a(of) p 408 2148 a(of) p 523 2148 a(The) p 683 2148 a(or) p 769 2148 a(em) p 927 2148 a(1.2.) p Fo 1163 2148 a(By) p 1316 2148 a(\(ii\)) p 1478 2148 a(of) p 1589 2148 a(Lemma) p 1937 2148 a(6.2,) p 2122 2148 a(w) m(e) p 2265 2148 a(ha) m(v) m(e) p Fn 1029 2384 a(A) p Fg 1102 2337 a(y) p Fm 1102 2408 a(N) p Fg 1170 2317 a(\003) p 1209 2281 141 4 v Fn 1209 2384 a(A) p Fg 1282 2337 a(y) p Fm 1282 2408 a(N) p Fi 1350 2259 a(\014) 1350 2309 y(\014) 1350 2359 y(\014) 1350 2409 y(\014) p Fl 1377 2463 a(\(Ker) p Fm 1530 2463 a(A) p Fb 1583 2430 a(y) p Fe 1583 2487 a(N) p Fb 1641 2416 a(\003) p 1677 2386 111 4 v Fm 1677 2463 a(A) p Fb 1730 2430 a(y) p Fe 1730 2487 a(N) p Fl 1787 2463 a(\)) p Fb 1814 2444 a(?) p Fk 1899 2384 a(') p 2004 2281 141 4 v Fn 2004 2384 a(A) p Fg 2077 2337 a(y) p Fm 2077 2408 a(N) p Fn 2144 2384 a(A) p Fg 2217 2337 a(y) p Fm 2217 2408 a(N) p Fg 2285 2317 a(\003) p Fi 2324 2259 a(\014) 2324 2309 y(\014) 2324 2359 y(\014) 2324 2409 y(\014) p Fl 2352 2463 a(\(Ker) p 2505 2386 111 4 v Fm 2505 2463 a(A) p Fb 2558 2430 a(y) p Fe 2558 2487 a(N) p Fm 2615 2463 a(A) p Fb 2668 2430 a(y) p Fe 2668 2487 a(N) p Fb 2726 2416 a(\003) p Fl 2762 2463 a(\)) p Fb 2789 2444 a(?) p Fn 2845 2384 a(;) p Fo 0 2649 a(where) p Fk 283 2649 a(') p Fo 394 2649 a(denotes) p 749 2649 a(the) p 918 2649 a(\(partial\)) p 1310 2649 a(unitarily) p 1709 2649 a(equiv) p 1931 2649 a(alence.) p 2266 2649 a(By) p 2420 2649 a(\(ii\)) p 2584 2649 a(and) p 2775 2649 a(\(iii\)) p 2966 2649 a(of) p 3079 2649 a(Prop) s(osition) p 3605 2649 a(6.3,) p 3791 2649 a(w) m(e) 0 2769 y(ha) m(v) m(e) p 225 2769 a(the) p 393 2769 a(unitarily) p 791 2769 a(equiv) p 1013 2769 a(alence) p 1306 2769 a(relation) 862 2968 y(\() p Fn(H) p Fm 989 2927 a(AB) 981 2992 y(N) p Fo 1125 2968 a(+) p Fn 1223 2968 a(B) p Fo 1302 2968 a(\)) p Fk(j) p Fl 1368 2990 a(\(Ker) o(\() p Fm(H) p Fe 1598 2968 a(AB) 1593 3015 y(N) p Fl 1698 2990 a(+) p Fm(B) p Fl 1809 2990 a(\)\)) p Fb 1863 2972 a(?) p Fk 1948 2968 a(') p Fo 2053 2968 a(\() p Fn(H) p Fg 2180 2927 a(\000) p Fm 2172 2992 a(N) p Fk 2261 2968 a(\000) p Fn 2361 2968 a(B) p Fo 2440 2968 a(\)) p Fk(j) p Fl 2506 2995 a(\(Ker\() p Fm(H) p Fb 2737 2965 a(\000) p Fe 2732 3020 a(N) p Fg 2790 2995 a(\000) p Fm(B) p Fl 2901 2995 a(\)\)) p Fb 2955 2976 a(?) p Fn 3012 2968 a(:) p Fo 3701 2968 a(\(7.2\)) 146 3181 y(Let) p Fn 321 3181 a(H) p Fm 402 3196 a(N) p Fo 502 3181 a(b) s(e) p 634 3181 a(an) m(y) p 819 3181 a(self-adjoin) m(t) p 1321 3181 a(extension) p 1752 3181 a(of) p Fn 1863 3181 a(L) p Fm 1929 3196 a(N) p Fo 1997 3181 a(.) p 2067 3181 a(By) p 2220 3181 a(\(i\)) p 2356 3181 a(of) p 2467 3181 a(Lemma) p 2815 3181 a(7.1,) p 2999 3181 a(w) m(e) p 3143 3181 a(ha) m(v) m(e) p Fn 1122 3380 a(S) p Fo 1215 3380 a(=) p Fn 1319 3380 a(\033) p Fm 1374 3395 a(ess) p Fo 1477 3380 a(\() p Fn(H) p Fm 1596 3395 a(N) p Fo 1663 3380 a(\)) p 1728 3380 a(=) p Fn 1832 3380 a(\033) p Fm 1887 3395 a(ess) p Fo 1989 3380 a(\() p Fn(H) p Fm 2116 3339 a(AB) 2108 3404 y(N) p Fo 2229 3380 a(\)) p 2295 3380 a(=) p Fn 2398 3380 a(\033) p Fm 2453 3395 a(ess) p Fo 2556 3380 a(\() p Fn(H) p Fg 2683 3339 a(\000) p Fm 2675 3404 a(N) p Fo 2742 3380 a(\)) 0 3578 y(for) p 150 3578 a(some) p 396 3578 a(closed) p 684 3578 a(subset) p Fn 985 3578 a(S) p Fk 1080 3578 a(\032) p Fj 1187 3578 a(R) p Fo 1308 3578 a(indep) s(enden) m(t) p 1862 3578 a(of) p 1974 3578 a(the) p 2143 3578 a(c) m(hoice) p 2434 3578 a(of) p 2546 3578 a(the) p 2715 3578 a(self-adjoin) m(t) p 3218 3578 a(extension) p Fn 3650 3578 a(H) p Fo 3739 3578 a(.) p 3813 3578 a(In) 0 3699 y(particular) p 454 3699 a(w) m(e) p 602 3699 a(ha) m(v) m(e) p Fn 831 3699 a(S) p Fk 932 3699 a(\032) p Fo 1044 3699 a([) p Fn(B) p 1150 3699 a(;) p Fo 1194 3699 a(+) p Fk(1) p Fo(\),) p 1473 3699 a(since) p Fn 1716 3699 a(H) p Fm 1805 3663 a(AB) 1797 3723 y(N) p Fk 1953 3699 a(\025) p Fn 2065 3699 a(B) p Fo 2181 3699 a(in) p 2299 3699 a(the) p 2471 3699 a(form) p 2706 3699 a(sense) p 2961 3699 a(b) m(y) p 3100 3699 a(\(iii\)) p 3294 3699 a(of) p 3410 3699 a(Prop) s(osition) 0 3819 y(5.7.) p 195 3819 a(Th) m(us) p Fk 442 3819 a(f\000) p Fn(B) p Fk 648 3819 a(g) p Fn 738 3819 a(=) p Fk 726 3819 a(2) p Fn 820 3819 a(S) p Fo 886 3819 a(.) p 956 3819 a(By) p 1110 3819 a(\(7.2\),) p 1370 3819 a(w) m(e) p 1513 3819 a(ha) m(v) m(e) p Fn 1135 4018 a(S) p Fk 1223 4018 a(n) p 1295 4018 a(f) p Fn(B) p Fk 1424 4018 a(g) p Fo 1501 4018 a(=) p Fn 1605 4018 a(S) p Fo 1693 4018 a(+) p 1791 4018 a(2) p Fn(B) p Fo 1946 4018 a(=) p Fk 2050 4018 a(f) p Fn(x) p Fo 2177 4018 a(+) p 2275 4018 a(2) p Fn(B) p Fo 2403 4018 a(;) p Fn 2447 4018 a(x) p Fk 2530 4018 a(2) p Fn 2624 4018 a(S) p Fk 2690 4018 a(g) p Fn(:) p Fo 3701 4018 a(\(7.3\)) 146 4217 y(It) p 246 4217 a(is) p 337 4217 a(easy) p 545 4217 a(to) p 658 4217 a(sho) m(w) p 894 4217 a(the) p 1055 4217 a(set) p 1201 4217 a(satisfying) p 1629 4217 a(\(7.3\)) p 1856 4217 a(is) p Fk 1948 4217 a(f) p Fo(\(2) p Fn(n) p Fk 2152 4217 a(\000) p Fo 2238 4217 a(1\)) p Fn(B) p Fo 2404 4217 a(;) p Fn 2448 4217 a(n) p Fo 2533 4217 a(=) p 2637 4217 a(1) p Fn(;) p Fo 2730 4217 a(2) p Fn(;) p 2823 4217 a(:) p 2867 4217 a(:) p 2911 4217 a(:) p Fk 2936 4217 a(g) p Fo 3012 4217 a(or) p 3125 4217 a(the) p 3287 4217 a(empt) m(y) p 3578 4217 a(set.) p 3767 4217 a(W) p 3859 4217 a(e) 0 4337 y(sho) m(w) p Fn 233 4337 a(S) p Fo 322 4337 a(is) p 412 4337 a(not) p 576 4337 a(empt) m(y) p 833 4337 a(.) p 901 4337 a(T) p 963 4337 a(o) p 1036 4337 a(see) p 1185 4337 a(this,) p 1395 4337 a(it) p 1483 4337 a(is) p 1573 4337 a(su\016cien) m(t) p 1974 4337 a(to) p 2084 4337 a(construct) p 2504 4337 a(a) p 2576 4337 a(W) p 2668 4337 a(eyl) p 2814 4337 a(sequence) p 3209 4337 a(for) p 3349 4337 a(the) p 3509 4337 a(sp) s(ectrum) p Fn 0 4458 a(B) p Fo 79 4458 a(,) p 139 4458 a(that) p 351 4458 a(is,) p 476 4458 a(an) p 612 4458 a(orthonormal) p 1170 4458 a(sequence) p Fk 1575 4458 a(f) p Fn(u) p Fm 1681 4473 a(n) p Fk 1727 4458 a(g) p Fg 1777 4421 a(1) p Fm 1777 4482 a(n) p Fl(=1) p Fo 1947 4458 a(suc) m(h) p 2167 4458 a(that) p Fk 2378 4458 a(jj) p Fo(\() p Fn(H) p Fk 2582 4458 a(\000) p Fn 2682 4458 a(B) p Fo 2761 4458 a(\)) p Fn(u) p Fm 2855 4473 a(n) p Fk 2902 4458 a(jj) p 2985 4458 a(!) p Fo 3113 4458 a(0) p 3194 4458 a(as) p Fn 3314 4458 a(n) p Fk 3400 4458 a(!) p 3528 4458 a(1) p Fo(.) p 3699 4458 a(T) p 3761 4458 a(ak) m(e) 0 4578 y(coun) m(table) p 445 4578 a(disjoin) m(t) p 795 4578 a(balls) p Fk 1026 4578 a(f) p Fn(B) p Fo 1155 4578 a(\() p Fn(n) p Fo(;) p Fn 1295 4578 a(w) p Fm 1365 4593 a(n) p Fo 1411 4578 a(\)) p Fk(g) p Fg 1499 4542 a(1) p Fm 1499 4603 a(n) p Fl(=1) p Fo 1636 4578 a(,) p Fn 1699 4578 a(B) p Fo 1778 4578 a(\() p Fn(n) p Fo(;) p Fn 1918 4578 a(w) p Fm 1988 4593 a(n) p Fo 2035 4578 a(\)) p 2105 4578 a(=) p Fk 2214 4578 a(f) p Fn(z) p Fo 2313 4578 a(;) p Fk 2357 4578 a(j) p Fn(z) p Fk 2459 4578 a(\000) p Fn 2560 4578 a(w) p Fm 2630 4593 a(n) p Fk 2677 4578 a(j) p Fn 2737 4578 a(<) p 2845 4578 a(n) p Fk(g) p Fo 2989 4578 a(con) m(tained) p 3433 4578 a(in) p Fj 3550 4578 a(R) p Fl 3637 4536 a(2) p Fk 3701 4578 a(n) p Fn 3775 4578 a(S) p Fm 3835 4593 a(N) p Fo 0 4698 a(and) p 190 4698 a(tak) m(e) p Fn 401 4698 a(\037) p Fk 490 4698 a(2) p Fn 584 4698 a(C) p Fg 661 4662 a(1) p Fl 654 4723 a(0) p Fo 736 4698 a(\() p Fk(fj) p Fn(z) p Fk 901 4698 a(j) p Fn 956 4698 a(<) p Fo 1060 4698 a(1) p Fk(g) p Fo(\)) p 1229 4698 a(with) p Fn 1451 4698 a(\037) p Fo(\() p Fn(z) p Fo 1599 4698 a(\)) p 1665 4698 a(=) p 1769 4698 a(1) p 1850 4698 a(in) p Fk 1964 4698 a(fj) p Fn(z) p Fk 2091 4698 a(j) p Fn 2147 4698 a(<) p Fl 2260 4659 a(1) p 2260 4675 36 4 v 2260 4733 a(2) p Fk 2305 4698 a(g) p Fo(.) p 2426 4698 a(Put) p Fn 1447 4930 a(v) p Fm 1494 4945 a(n) p Fo 1541 4930 a(\() p Fn(z) p Fo 1628 4930 a(\)) p 1694 4930 a(=) p Fn 1798 4930 a(t) p Fm 1833 4945 a(w) p Fe 1884 4953 a(n) p Fn 1931 4930 a(\037) p Fo(\() p Fn 2044 4862 a(z) p 2040 4906 59 4 v 2040 4998 a(n) p Fo 2108 4930 a(\)) p Fn(e) p Fg 2191 4888 a(\000) p Fe 2256 4861 a(B) p 2256 4873 49 4 v Fd 2265 4914 a(4) p Fg 2314 4888 a(j) p Fm(z) p Fg 2370 4888 a(j) p Fd 2390 4865 a(2) p Fn 2428 4930 a(;) p Fo 1902 5214 a(43) p eop %%Page: 44 44 44 43 bop Fo 0 631 a(where) p Fn 279 631 a(t) p Fm 314 646 a(w) p Fe 365 654 a(n) p Fo 442 631 a(is) p 537 631 a(the) p 703 631 a(magnetic) p 1117 631 a(translation) p 1609 631 a(from) p Fk 1836 631 a(fj) p Fn(z) p Fk 1963 631 a(j) p Fn 2019 631 a(<) p 2123 631 a(n) p Fk(g) p Fo 2261 631 a(to) p Fk 2377 631 a(fj) p Fn(z) p Fk 2521 631 a(\000) p Fn 2616 631 a(w) p Fm 2686 646 a(n) p Fk 2732 631 a(j) p Fn 2788 631 a(<) p 2891 631 a(n) p Fk(g) p Fo 3029 631 a(in) m(tert) m(wining) p Fk 3574 631 a(L) p Fl 3643 646 a(0) p Fo 3712 631 a(with) p Fk 0 751 a(L) p Fm 69 766 a(N) p Fo 136 751 a(.) p 206 751 a(It) p 312 751 a(is) p 410 751 a(easy) p 625 751 a(to) p 744 751 a(c) m(hec) m(k) p 1007 751 a(that) p Fk 1218 751 a(f) p Fn(v) p Fm 1315 766 a(n) p Fn 1362 751 a(=) p Fk(jj) p Fn(v) p Fm 1514 766 a(n) p Fk 1560 751 a(jjg) p Fg 1666 715 a(1) p Fm 1666 776 a(n) p Fl(=1) p Fo 1835 751 a(is) p 1933 751 a(the) p 2101 751 a(W) p 2193 751 a(eyl) p 2348 751 a(sequence) p 2752 751 a(b) m(y) p 2888 751 a(using) p 3143 751 a(the) p 3311 751 a(equalit) m(y) p Fk 1036 964 a(L) p Fm 1105 979 a(N) p Fn 1172 964 a(t) p Fm 1207 979 a(w) p Fe 1258 987 a(n) p Fn 1305 964 a(e) p Fg 1350 923 a(\000) p Fe 1415 896 a(B) p 1415 908 49 4 v Fd 1424 949 a(4) p Fg 1473 923 a(j) p Fm(z) p Fg 1529 923 a(j) p Fd 1549 899 a(2) p Fo 1614 964 a(=) p Fn 1718 964 a(t) p Fm 1753 979 a(w) p Fe 1804 987 a(n) p Fk 1851 964 a(L) p Fl 1920 979 a(0) p Fn 1959 964 a(e) p Fg 2004 923 a(\000) p Fe 2069 896 a(B) p 2069 908 V Fd 2078 949 a(4) p Fg 2127 923 a(j) p Fm(z) p Fg 2183 923 a(j) p Fd 2203 899 a(2) p Fo 2268 964 a(=) p Fn 2372 964 a(B) p 2451 964 a(t) p Fm 2486 979 a(w) p Fe 2537 987 a(n) p Fn 2584 964 a(e) p Fg 2629 923 a(\000) p Fe 2694 896 a(B) p 2694 908 V Fd 2703 949 a(4) p Fg 2752 923 a(j) p Fm(z) p Fg 2808 923 a(j) p Fd 2828 899 a(2) p Fo 0 1162 a(and) p 190 1162 a(the) p 358 1162 a(Leibniz) p 700 1162 a(rule.) p 933 1162 a(Th) m(us) p 1181 1162 a(w) m(e) p 1324 1162 a(ha) m(v) m(e) p Fn 1146 1359 a(\033) p Fm 1201 1374 a(ess) p Fo 1304 1359 a(\() p Fn(H) p Fm 1423 1374 a(N) p Fo 1490 1359 a(\)) p 1555 1359 a(=) p Fk 1659 1359 a(f) p Fo(\(2) p Fn(n) p Fk 1876 1359 a(\000) p Fo 1975 1359 a(1\)) p Fn(B) p Fo 2141 1359 a(;) p Fn 2217 1359 a(n) p Fo 2303 1359 a(=) p 2407 1359 a(1) p Fn(;) p Fo 2500 1359 a(2) p Fn(;) p 2593 1359 a(:) p 2637 1359 a(:) p 2681 1359 a(:) p Fk 2706 1359 a(g) p Fo 3701 1359 a(\(7.4\)) 0 1557 y(for) p 149 1557 a(an) m(y) p 333 1557 a(self-adjoin) m(t) p 835 1557 a(extension) p Fn 1267 1557 a(H) p Fm 1348 1572 a(N) p Fo 1447 1557 a(of) p Fn 1558 1557 a(L) p Fm 1624 1572 a(N) p Fo 1692 1557 a(.) 146 1677 y(Next,) p 412 1677 a(w) m(e) p 555 1677 a(shall) p 784 1677 a(pro) m(v) m(e) p 1046 1677 a(\(ii\).) p 1247 1677 a(The) p 1447 1677 a(assertion) 1375 1875 y(dim) p 1554 1875 a(Ran) p Fn 1746 1875 a(P) p Fl 1809 1890 a(\() p Fg(\0001) p Fm(;B) p Fl 2038 1890 a(\)) p Fo 2069 1875 a(\() p Fn(H) p Fm 2196 1833 a(AB) 2188 1899 y(N) p Fo 2309 1875 a(\)) p 2375 1875 a(=) p 2478 1875 a(0) p 3701 1875 a(\(7.5\)) 0 2072 y(follo) m(ws) p 320 2072 a(from) p 551 2072 a(\(iii\)) p 740 2072 a(of) p 851 2072 a(Prop) s(osition) p 1376 2072 a(5.7.) p 1571 2072 a(By) p 1724 2072 a(the) p 1892 2072 a(unitarily) p 2291 2072 a(equiv) p 2513 2072 a(alence) p 2806 2072 a(relation) p 3163 2072 a(\(7.2\),) p 3423 2072 a(w) m(e) p 3567 2072 a(ha) m(v) m(e) 563 2270 y(dim) p 742 2270 a(Ran) p Fn 934 2270 a(P) p Fl 997 2285 a(\(\(2) p Fm(n) p Fg(\000) p Fl(1\)) p Fm(B) s(;) p Fl(\(2) p Fm(n) p Fl(+1\)) p Fm(B) p Fl 1600 2285 a(\)) p Fo 1634 2270 a(\() p Fn(H) p Fm 1761 2228 a(AB) 1753 2294 y(N) p Fo 1874 2270 a(\)) p 1940 2270 a(=) p 2044 2270 a(dim) p 2223 2270 a(Ran) p Fn 2414 2270 a(P) p Fl 2477 2285 a(\(\(2) p Fm(n) p Fl(+1\)) p Fm(B) s(;) p Fl(\(2) p Fm(n) p Fl(+3\)) p Fm(B) p Fl 3080 2285 a(\)) p Fo 3115 2270 a(\() p Fn(H) p Fg 3242 2228 a(\000) p Fm 3234 2294 a(N) p Fo 3301 2270 a(\)) p 3701 2270 a(\(7.6\)) 0 2467 y(for) p 149 2467 a(ev) m(ery) p Fn 406 2467 a(n) p Fo 492 2467 a(=) p 596 2467 a(0) p Fn(;) p Fo 689 2467 a(1) p Fn(;) p Fo 782 2467 a(2) p Fn(;) p 875 2467 a(:) p 919 2467 a(:) p 963 2467 a(:) p Fo 988 2467 a(.) p 1058 2467 a(By) p 1211 2467 a(\(iv\)) p 1398 2467 a(of) p 1509 2467 a(Prop) s(osition) p 2034 2467 a(6.3) p 2191 2467 a(and) p 2381 2467 a(\(ii\)) p 2544 2467 a(of) p 2655 2467 a(Lemma) p 3003 2467 a(7.1,) p 3187 2467 a(w) m(e) p 3331 2467 a(ha) m(v) m(e) 458 2665 y(dim) p 637 2665 a(Ran) p Fn 828 2665 a(P) p Fl 891 2680 a(\(\(2) p Fm(n) p Fg(\000) p Fl(1\)) p Fm(B) s(;) p Fl(\(2) p Fm(n) p Fl(+1\)) p Fm(B) p Fl 1494 2680 a(\)) p Fo 1529 2665 a(\() p Fn(H) p Fg 1656 2623 a(\000) p Fm 1648 2689 a(N) p Fo 1715 2665 a(\)) p Fk 1781 2665 a(\024) p Fo 1886 2665 a(dim) p 2065 2665 a(Ran) p Fn 2257 2665 a(P) p Fl 2320 2680 a(\(\(2) p Fm(n) p Fg(\000) p Fl(1\)) p Fm(B) s(;) p Fl(\(2) p Fm(n) p Fl(+1\)) p Fm(B) p Fl 2923 2680 a(\)) p Fo 2957 2665 a(\() p Fn(H) p Fm 3084 2623 a(AB) 3076 2689 y(N) p Fo 3197 2665 a(\)) p 3258 2665 a(+) p Fn 3356 2665 a(N) p Fo 3701 2665 a(\(7.7\)) 0 2862 y(for) p 155 2862 a(ev) m(ery) p Fn 418 2862 a(n) p Fo 514 2862 a(=) p 627 2862 a(0) p Fn(;) p Fo 720 2862 a(1) p Fn(;) p Fo 813 2862 a(2) p Fn(;) p 906 2862 a(:) p 950 2862 a(:) p 994 2862 a(:) p Fo 1019 2862 a(.) p 1107 2862 a(By) p 1266 2862 a(\(7.5\),) p 1533 2862 a(\(7.6\),) p 1801 2862 a(\(7.7\)) p 2040 2862 a(and) p 2235 2862 a(an) p 2376 2862 a(elemen) m(tary) p 2881 2862 a(induction) p 3320 2862 a(argumen) m(t,) p 3791 2862 a(w) m(e) 0 2983 y(ha) m(v) m(e) p 225 2983 a(\(1.9\).) p 496 2983 a(Th) m(us) p 743 2983 a(\(ii\)) p 906 2983 a(of) p 1017 2983 a(Theorem) p 1429 2983 a(1.2) p 1586 2983 a(holds.) 146 3103 y(Since) p 401 3103 a(the) p 569 3103 a(de\014ciency) p 1016 3103 a(indices) p 1336 3103 a(of) p Fn 1448 3103 a(L) p Fm 1514 3118 a(N) p Fo 1614 3103 a(are) p 1776 3103 a(\(2) p Fn(N) p 1946 3103 a(;) p Fo 1990 3103 a(2) p Fn(N) p Fo 2127 3103 a(\),) p 2225 3103 a(w) m(e) p 2368 3103 a(ha) m(v) m(e) 1195 3313 y(dim) p Fi 1374 3217 a(\020) p Fn 1424 3313 a(D) p Fo 1508 3313 a(\() p Fn(H) p Fo 1635 3313 a(\)) p Fn(=D) p Fo 1806 3313 a(\() p Fn(H) p Fo 1933 3313 a(\)) p Fk 1991 3313 a(\\) p Fn 2080 3313 a(D) p Fo 2164 3313 a(\() p Fn(H) p Fg 2291 3272 a(\000) p Fm 2283 3338 a(N) p Fo 2349 3313 a(\)) p Fi 2387 3217 a(\021) p Fk 2464 3313 a(\024) p Fo 2570 3313 a(2) p Fn(N) p Fo 3701 3313 a(\(7.8\)) 0 3523 y(for) p 148 3523 a(an) m(y) p 331 3523 a(self-adjoin) m(t) p 832 3523 a(extension) p Fn 1262 3523 a(H) p Fm 1343 3538 a(N) p Fo 1442 3523 a(of) p Fn 1552 3523 a(L) p Fm 1618 3538 a(N) p Fo 1685 3523 a(.) p 1755 3523 a(Th) m(us) p 2001 3523 a(\(iii\)) p 2190 3523 a(of) p 2300 3523 a(Theorem) p 2711 3523 a(1.2) p 2867 3523 a(follo) m(ws) p 3186 3523 a(from) p 3415 3523 a(\(1.9\),) p 3674 3523 a(\(7.6\),) 0 3643 y(\(7.8\)) p 233 3643 a(and) p 423 3643 a(\(ii\)) p 585 3643 a(of) p 696 3643 a(Lemma) p 1045 3643 a(7.1.) 146 3764 y(T) p 208 3764 a(o) p 290 3764 a(sho) m(w) p 532 3764 a(\(i\)) p 667 3764 a(of) p 778 3764 a(Theorem) p 1190 3764 a(1.2,) p 1374 3764 a(consider) p 1754 3764 a(the) p 1922 3764 a(follo) m(wing) p 2334 3764 a(equalit) m(y:) 620 3961 y(dim) p 799 3961 a(Ran) p Fn 990 3961 a(P) p Fg 1053 3977 a(f) p Fl(\(2) p Fm(n) p Fg(\000) p Fl(1\)) p Fm(B) p Fg 1366 3977 a(g) p Fo 1407 3961 a(\() p Fn(H) p Fm 1526 3976 a(N) p Fo 1593 3961 a(\)) 460 4106 y(=) p 620 4106 a(dim) p 799 4106 a(Ran) p Fn 990 4106 a(P) p Fl 1053 4122 a(\(\(2) p Fm(n) p Fg(\000) p Fl(1\)) p Fm(B) p Fg 1358 4122 a(\000) p Fm(\017;) p Fl(\(2) p Fm(n) p Fg(\000) p Fl(1\)) p Fm(B) p Fl 1740 4122 a(+) p Fm(\017) p Fl(\)) p Fo 1858 4106 a(\() p Fn(H) p Fm 1977 4121 a(N) p Fo 2044 4106 a(\)) p Fk 459 4252 a(\000) p Fo 620 4252 a(dim) p 799 4252 a(Ran) p Fn 990 4252 a(P) p Fl 1053 4267 a(\(\(2) p Fm(n) p Fg(\000) p Fl(1\)) p Fm(B) p Fg 1358 4267 a(\000) p Fm(\017;) p Fl(\(2) p Fm(n) p Fg(\000) p Fl(1\)) p Fm(B) p Fl 1740 4267 a(\)) p Fo 1774 4252 a(\() p Fn(H) p Fm 1893 4267 a(N) p Fo 1960 4252 a(\)) p Fk 2020 4252 a(\000) p Fo 2120 4252 a(dim) p 2299 4252 a(Ran) p Fn 2491 4252 a(P) p Fl 2554 4267 a(\(\(2) p Fm(n) p Fg(\000) p Fl(1\)) p Fm(B) s(;) p Fl(\(2) p Fm(n) p Fg(\000) p Fl(1\)) p Fm(B) p Fl 3157 4267 a(+) p Fm(\017) p Fl(\)) p Fo 3275 4252 a(\() p Fn(H) p Fm 3394 4267 a(N) p Fo 3461 4252 a(\)) p Fn(;) p Fo 0 4449 a(The) p 199 4449 a(\014rst) p 398 4449 a(term) p 630 4449 a(of) p 739 4449 a(the) p 906 4449 a(last) p 1089 4449 a(expression) p 1557 4449 a(is) p 1654 4449 a(in\014nit) m(y) p 1988 4449 a(b) m(y) p 2122 4449 a(\(7.4\).) p 2393 4449 a(The) p 2592 4449 a(second) p 2905 4449 a(and) p 3093 4449 a(the) p 3260 4449 a(third) p 3502 4449 a(terms) p 3772 4449 a(are) 0 4570 y(\014nite) p 249 4570 a(n) m(um) m(b) s(ers) p 643 4570 a(for) p 792 4570 a(small) p Fn 1047 4570 a(\017) p 1114 4570 a(>) p Fo 1218 4570 a(0,) p 1326 4570 a(b) m(y) p 1462 4570 a(\(iii\)) p 1651 4570 a(of) p 1762 4570 a(Theorem) p 2174 4570 a(1.2.) p 2369 4570 a(Therefore) p 2811 4570 a(w) m(e) p 2955 4570 a(ha) m(v) m(e) 1428 4767 y(dim) p 1607 4767 a(Ran) p Fn 1799 4767 a(P) p Fg 1862 4783 a(f) p Fl(\(2) p Fm(n) p Fg(\000) p Fl(1\)) p Fm(B) p Fg 2175 4783 a(g) p Fo 2244 4767 a(=) p Fk 2347 4767 a(1) p Fn(;) p Fo 0 4965 a(whic) m(h) p 279 4965 a(implies) p 610 4965 a(\(i\)) p 746 4965 a(of) p 857 4965 a(Theorem) p 1269 4965 a(1.2.) p 1464 4897 64 4 v 1464 4990 4 93 v 1524 4990 V 1464 4993 64 4 v 1902 5214 a(44) p eop %%Page: 45 45 45 44 bop Fo 0 631 a(Remark.) p 431 631 a(If) p 536 631 a(w) m(e) p 689 631 a(apply) p 965 631 a(the) p 1142 631 a(argumen) m(t) p 1587 631 a(in) p 1709 631 a(pro) m(ving) p 2070 631 a(\(7.6\)) p 2311 631 a(to) p 2439 631 a(the) p 2615 631 a(pair) p 2824 631 a(of) p 2944 631 a(op) s(erators) p 3384 553 141 4 v Fn 3384 631 a(A) p Fm 3457 646 a(N) p Fn 3524 631 a(A) p Fm 3597 646 a(N) p Fg 3664 589 a(\003) p Fo 3745 631 a(and) p Fn 0 751 a(A) p Fm 73 766 a(N) p Fg 140 710 a(\003) p 180 673 V Fn 180 751 a(A) p Fm 253 766 a(N) p Fo 320 751 a(,) p 380 751 a(w) m(e) p 523 751 a(ha) m(v) m(e) 563 933 y(dim) p 742 933 a(Ran) p Fn 934 933 a(P) p Fl 997 948 a(\(\(2) p Fm(n) p Fg(\000) p Fl(1\)) p Fm(B) s(;) p Fl(\(2) p Fm(n) p Fl(+1\)) p Fm(B) p Fl 1600 948 a(\)) p Fo 1634 933 a(\() p Fn(H) p Fm 1761 892 a(AB) 1753 958 y(N) p Fo 1874 933 a(\)) p 1940 933 a(=) p 2044 933 a(dim) p 2223 933 a(Ran) p Fn 2414 933 a(P) p Fl 2477 948 a(\(\(2) p Fm(n) p Fg(\000) p Fl(3\)) p Fm(B) s(;) p Fl(\(2) p Fm(n) p Fg(\000) p Fl(1\)) p Fm(B) p Fl 3080 948 a(\)) p Fo 3115 933 a(\() p Fn(H) p Fl 3242 892 a(+) p Fm 3234 958 a(N) p Fo 3301 933 a(\)) 0 1115 y(for) p Fn 149 1115 a(n) p Fo 235 1115 a(=) p 338 1115 a(1) p Fn(;) p Fo 431 1115 a(2) p Fn(;) p 524 1115 a(:) p 568 1115 a(:) p 612 1115 a(:) p Fo 638 1115 a(.) p 708 1115 a(When) p Fn 993 1115 a(N) p Fo 1109 1115 a(=) p 1213 1115 a(1,) p 1321 1115 a(this) p 1511 1115 a(equalit) m(y) p 1882 1115 a(and) p 2072 1115 a(\(7.6\)) p 2305 1115 a(imply) 1068 1296 y(dim) p 1247 1296 a(Ran) p Fn 1439 1296 a(P) p Fl 1502 1312 a(\(\(2) p Fm(n) p Fg(\000) p Fl(1\)) p Fm(B) s(;) p Fl(\(2) p Fm(n) p Fl(+1\)) p Fm(B) p Fl 2105 1312 a(\)) p Fo 2140 1296 a(\() p Fn(H) p Fl 2267 1255 a(+) 2259 1321 y(1) p Fo 2325 1296 a(\)) p 2446 1296 a(=) p Fn 2605 1296 a(n) p Fo 2685 1296 a(+) p 2783 1296 a(1) 1068 1441 y(dim) p 1247 1441 a(Ran) p 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t(of) p 687 2071 a(of) p 871 2071 a(Theorem) p 1555 2071 a(1.3) p Fo 0 2290 a(In) p 127 2290 a(this) p 322 2290 a(section,) p 681 2290 a(w) m(e) p 830 2290 a(shall) p 1063 2290 a(pro) m(v) m(e) p 1331 2290 a(Theorem) p 1748 2290 a(1.3.) p 1958 2290 a(Before) p 2268 2290 a(the) p 2441 2290 a(pro) s(of,) p 2729 2290 a(w) m(e) p 2878 2290 a(shall) p 3111 2290 a(mak) m(e) p 3371 2290 a(sure) p 3583 2290 a(that) p 3799 2290 a(an) 0 2410 y(op) s(erator) p 393 2410 a(satisfying) p 828 2410 a(the) p 996 2410 a(assumption) p 1511 2410 a(of) p 1623 2410 a(Theorem) p 2034 2410 a(1.3) p 2192 2410 a(exists.) p Fq 146 2580 a(Lemma) p 543 2580 a(8.1) p Ff 735 2580 a(L) p 791 2580 a(et) p Fn 905 2580 a(N) p Fo 1024 2580 a(=) p 1131 2580 a(1) p Fn(;) p Fo 1224 2580 a(2) p Fn(;) p 1317 2580 a(:) p 1361 2580 a(:) p 1405 2580 a(:) p Ff 1467 2580 a(or) p Fn 1594 2580 a(N) p Fo 1713 2580 a(=) p Fk 1819 2580 a(1) p Ff(.) p 1998 2580 a(L) p 2054 2580 a(et) p Fo 2168 2580 a(0) p Fn 2247 2580 a(<) p 2354 2580 a(\013) p 2447 2580 a(<) p Fo 2553 2580 a(1) p Ff 2639 2580 a(and) p 2830 2580 a(supp) p 3017 2580 a(ose) p Fn 3187 2580 a(\013) p Fm 3249 2595 a(j) p Fo 3316 2580 a(=) p Fn 3423 2580 a(\013) p Ff 3522 2580 a(for) p 3679 2580 a(every) p Fn 0 2701 a(j) p Fo 89 2701 a(=) p 208 2701 a(1) p Fn(;) p Fo 301 2701 a(2) p Fn(;) p 394 2701 a(:) p 438 2701 a(:) p 482 2701 a(:) p 524 2701 a(;) p 568 2701 a(N) p Ff 656 2701 a(.) p 756 2701 a(L) p 812 2701 a(et) p Fn 932 2701 a(H) p Fl 1013 2716 a(1) p Ff 1096 2701 a(b) p 1136 2701 a(e) p 1223 2701 a(any) p 1419 2701 a(self-adjoint) p 1932 2701 a(extension) p 2372 2701 a(of) p Fn 2494 2701 a(L) p Fm 2560 2665 a(\013) p Fl 2560 2725 a(1) p Ff 2610 2701 a(.) p 2710 2701 a(Then,) p 3004 2701 a(ther) p 3167 2701 a(e) p 3255 2701 a(exists) p 3530 2701 a(a) p 3623 2701 a(unique) 0 2821 y(self-adjoint) p 506 2821 a(extension) p Fn 936 2821 a(H) p Fm 1017 2836 a(N) p Ff 1119 2821 a(of) p Fn 1234 2821 a(L) p Fm 1300 2836 a(N) p Ff 1402 2821 a(satisfying) p 1836 2821 a(\(1.14\).) 146 3006 y(Pr) p 248 3006 a(o) p 293 3006 a(of.) p Fo 435 3006 a(By) p 589 3006 a(Corollary) p 1019 3006 a(5.5,) p 1204 3006 a(there) p 1453 3006 a(exists) p 1722 3006 a(a) p 1804 3006 a(4) p Fk 1875 3006 a(\002) p Fo 1974 3006 a(4) p 2055 3006 a(orthogonal) p 2543 3006 a(pro) p 2689 3006 a(jection) p 3007 3006 a(matrix) p Fn 3324 3006 a(M) p Fl 3418 3021 a(1) p Fo 3490 3006 a(satisfying) 1140 3187 y(rank) p Fn(M) p Fl 1426 3202 a(1) p Fo 1493 3187 a(=) p 1597 3187 a(2) p Fn(;) p 1722 3187 a(M) p Fl 1816 3202 a(1) p Fn 1856 3187 a(J) p Fm 1910 3202 a(\013) p Fn 1959 3187 a(M) p Fl 2053 3202 a(1) p Fo 2121 3187 a(=) p Fn 2224 3187 a(O) s(;) 1140 3333 y(D) p Fo 1224 3333 a(\() p Fn(H) p Fl 1343 3348 a(1) p Fo 1381 3333 a(\)) p 1447 3333 a(=) p Fk 1551 3333 a(f) p Fn(u) p Fk 1684 3333 a(2) p Fn 1778 3333 a(D) p Fo 1862 3333 a(\(\() p Fn(L) p Fm 2004 3291 a(\013) p Fl 2004 3357 a(1) p Fo 2053 3333 a(\)) p Fg 2091 3291 a(\003) p Fo 2130 3333 a(\);) p 2245 3333 a(\004) p Fl 2310 3348 a(1) p Fn 2349 3333 a(u) p Fk 2432 3333 a(2) p Fo 2526 3333 a(Ran) p Fn 2718 3333 a(M) p Fl 2812 3348 a(1) p Fk 2852 3333 a(g) p Fn(;) p Fo 3701 3333 a(\(8.1\)) 0 3514 y(where) p 282 3514 a(\004) p Fl 347 3529 a(1) p Fo 419 3514 a(b) s(e) p 552 3514 a(the) p 720 3514 a(map) p 936 3514 a(\004) p 1034 3514 a(asso) s(ciated) p 1498 3514 a(to) p 1618 3514 a(the) p 1786 3514 a(op) s(erator) p Fn 2178 3514 a(L) p Fm 2244 3478 a(\013) p Fl 2244 3539 a(1) p Fo 2294 3514 a(.) p 2365 3514 a(Put) p Fn 1286 3874 a(M) p Fm 1380 3889 a(N) p Fo 1475 3874 a(=) p Fi 1579 3628 a(0) 1579 3774 y(B) 1579 3824 y(B) 1579 3874 y(B) 1579 3927 y(@) p Fn 1693 3693 a(M) p Fl 1787 3708 a(1) p Fn 1938 3693 a(O) p 2127 3693 a(:) p 2171 3693 a(:) p 2215 3693 a(:) p 2353 3693 a(O) 1721 3813 y(O) p 1910 3813 a(M) p Fl 2004 3828 a(1) p Fn 2127 3813 a(:) p 2171 3813 a(:) p 2215 3813 a(:) p 2353 3813 a(O) p Fo 1719 3933 a(.) p 1762 3933 a(.) p 1805 3933 a(.) p 1847 3933 a(.) p 1890 3933 a(.) p 1933 3933 a(.) p 1976 3933 a(.) p 2019 3933 a(.) p 2062 3933 a(.) p 2105 3933 a(.) p 2148 3933 a(.) p 2191 3933 a(.) p 2234 3933 a(.) p 2277 3933 a(.) p 2320 3933 a(.) p 2363 3933 a(.) p 2405 3933 a(.) p Fn 1721 4054 a(O) p 1938 4054 a(O) p 2127 4054 a(:) p 2171 4054 a(:) p 2215 4054 a(:) p 2324 4054 a(M) p Fl 2418 4069 a(1) p Fi 2500 3628 a(1) 2500 3774 y(C) 2500 3824 y(C) 2500 3874 y(C) 2500 3927 y(A) p Fn 2589 3874 a(:) p Fo 0 4250 a(Then) p Fn 260 4250 a(M) p Fm 354 4265 a(N) p Fo 459 4250 a(satis\014es) p 824 4250 a(\(5.24\),) p 1139 4250 a(and) p 1334 4250 a(th) m(us) p 1554 4250 a(satis\014es) p 1919 4250 a(\(5.18\)) p 2206 4250 a(b) m(y) p 2347 4250 a(Lemma) p 2700 4250 a(5.6.) p 2911 4250 a(Let) p Fn 3091 4250 a(H) p Fm 3180 4205 a(M) p Fe 3248 4216 a(N) p Fm 3172 4275 a(N) p Fo 3347 4250 a(b) s(e) p 3485 4250 a(the) p 3658 4250 a(corre-) 0 4371 y(sp) s(onding) p 412 4371 a(self-adjoin) m(t) p 911 4371 a(extension) p 1339 4371 a(giv) m(en) p 1591 4371 a(in) p 1702 4371 a(Prop) s(osition) p 2223 4371 a(5.4.) p 2418 4371 a(Let) p 2589 4371 a(\004) p Fm 2654 4386 a(N) p Fo 2751 4371 a(b) s(e) p 2881 4371 a(the) p 3046 4371 a(map) p 3259 4371 a(\004) p 3354 4371 a(asso) s(ciated) p 3815 4371 a(to) 0 4491 y(the) p 168 4491 a(op) s(erator) p Fn 561 4491 a(L) p Fm 627 4506 a(N) p Fo 695 4491 a(.) p 765 4491 a(By) p 918 4491 a(the) p 1086 4491 a(form) p 1316 4491 a(of) p 1427 4491 a(the) p 1595 4491 a(matrix) p Fn 1913 4491 a(M) p Fm 2007 4506 a(N) p Fo 2074 4491 a(,) p 2134 4491 a(w) m(e) p 2277 4491 a(ha) m(v) m(e) p 2502 4491 a(an) p 2638 4491 a(equiv) p 2860 4491 a(alence) p 3153 4491 a(relation) p Fn 620 4673 a(u) p Fk 703 4673 a(2) p Fn 797 4673 a(D) p Fo 881 4673 a(\() p Fn(H) p Fm 1008 4627 a(M) p Fe 1076 4638 a(N) p Fm 1000 4697 a(N) p Fo 1137 4673 a(\)) p Fk 1258 4673 a(,) p Fo 1440 4673 a(\004) p Fm 1505 4688 a(N) p Fn 1572 4673 a(u) p Fk 1656 4673 a(2) p Fo 1750 4673 a(Ran) p Fn 1941 4673 a(M) p Fm 2035 4688 a(N) p Fk 1258 4818 a(,) p Fo 1440 4818 a(\004) p Fl 1505 4833 a(1) p Fo 1545 4818 a(\() p Fn(T) p Fm 1640 4833 a(j) p Fn 1676 4818 a(u) p Fo(\)) p Fk 1797 4818 a(2) p Fo 1892 4818 a(Ran) p Fn 2083 4818 a(M) p Fl 2177 4833 a(1) p Fo 2249 4818 a(for) p 2398 4818 a(ev) m(ery) p Fn 2657 4818 a(j) p Fo 2731 4818 a(=) p 2834 4818 a(1) p Fn(;) p Fo 2927 4818 a(2) p Fn(;) p 3020 4818 a(:) p 3064 4818 a(:) p 3108 4818 a(:) p 3150 4818 a(;) p 3194 4818 a(N) p Fk 1258 4963 a(,) p Fn 1440 4963 a(t) p Fg 1475 4978 a(\000) p Fm(z) p Fe 1563 4988 a(j) p Fn 1600 4963 a(\037) p Fm 1661 4978 a(j) p Fn 1697 4963 a(u) p Fk 1781 4963 a(2) p Fn 1875 4963 a(D) p Fo 1959 4963 a(\() p Fn(H) p Fl 2078 4978 a(1) p Fo 2117 4963 a(\)) p 2187 4963 a(for) p 2336 4963 a(ev) m(ery) p Fn 2595 4963 a(j) p Fo 2669 4963 a(=) p 2772 4963 a(1) p Fn(;) p Fo 2865 4963 a(2) p Fn(;) p 2958 4963 a(:) p 3002 4963 a(:) p 3046 4963 a(:) p 3088 4963 a(;) p 3132 4963 a(N) p 3215 4963 a(:) p Fo 3701 4963 a(\(8.2\)) 1902 5214 y(45) p eop %%Page: 46 46 46 45 bop Fo 0 631 a(Th) m(us) p Fn 247 631 a(H) p Fm 336 585 a(M) p Fe 404 596 a(N) p Fm 328 655 a(N) p Fo 498 631 a(satis\014es) p 857 631 a(\(1.14\).) p 1177 631 a(The) p 1377 631 a(equiv) p 1599 631 a(alence) p 1892 631 a(\(8.2\)) p 2125 631 a(also) p 2321 631 a(implies) p 2652 631 a(the) p 2820 631 a(uniqueness.) p 3349 563 64 4 v 3349 656 4 93 v 3410 656 V 3349 659 64 4 v 146 859 a(In) p 267 859 a(the) p 434 859 a(sequel,) p Fn 751 859 a(H) p Fm 832 874 a(N) p Fo 931 859 a(denotes) p 1283 859 a(the) p 1450 859 a(op) s(erator) p 1842 859 a(giv) m(en) p 2095 859 a(in) p 2208 859 a(Lemma) p 2556 859 a(8.1.) p 2750 859 a(W) p 2842 859 a(e) p 2917 859 a(shall) p 3145 859 a(giv) m(e) p 3344 859 a(some) p 3588 859 a(domain) 0 980 y(prop) s(erties) p 458 980 a(of) p Fn 570 980 a(H) p Fm 651 995 a(N) p Fo 718 980 a(.) p Fq 146 1183 a(Lemma) p 543 1183 a(8.2) p Ff 735 1183 a(L) p 791 1183 a(et) p Fn 908 1183 a(\021) p Fk 996 1183 a(2) p Fn 1099 1183 a(C) p Fg 1176 1147 a(1) p Fl 1169 1208 a(0) p Fo 1250 1183 a(\() p Fk(fj) p Fn(z) p Fk 1415 1183 a(j) p Fn 1480 1183 a(<) p Fm 1602 1144 a(R) p 1602 1160 54 4 v Fl 1611 1217 a(2) p Fk 1665 1183 a(g) p Fo(\)) p Ff 1792 1183 a(with) p Fn 2009 1183 a(\021) p Fo 2097 1183 a(=) p 2209 1183 a(1) p Ff 2297 1183 a(on) p 2441 1183 a(some) p 2695 1183 a(op) p 2790 1183 a(en) p 2928 1183 a(neighb) p 3193 1183 a(orho) p 3379 1183 a(o) p 3424 1183 a(d) p 3512 1183 a(of) p Fo 3631 1183 a(0) p Ff(.) p 3769 1183 a(L) p 3825 1183 a(et) p Fo 0 1303 a(1) p Fk 76 1303 a(\024) p Fn 182 1303 a(j) p Fk 255 1303 a(\024) p Fn 360 1303 a(N) p Ff 477 1303 a(and) p 660 1303 a(put) p Fn 822 1303 a(\021) p Fm 870 1318 a(j) p Fo 907 1303 a(\() p Fn(z) p Fo 994 1303 a(\)) p 1060 1303 a(=) p Fn 1164 1303 a(\021) p Fo 1216 1303 a(\() p Fn(z) p Fk 1310 1303 a(\000) p Fn 1394 1303 a(z) p Fm 1439 1318 a(j) p Fo 1476 1303 a(\)) p Ff(.) p 1587 1303 a(L) p 1643 1303 a(et) p Fn 1748 1303 a(t) p Fg 1783 1318 a(\000) p Fm(z) p Fe 1871 1328 a(j) p Ff 1936 1303 a(b) p 1976 1303 a(e) p 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p Fl 2230 2340 a(1) p Fo 2269 2325 a(\() p Fn(\021) t(v) p Fo 2410 2325 a(\)) p Fn(:) p Fo 3701 2325 a(\(8.4\)) p Ff 146 2553 a(Pr) p 248 2553 a(o) p 293 2553 a(of.) p Fo 447 2553 a(T) p 509 2553 a(ak) m(e) p Fn 694 2553 a(u) p Fk 797 2553 a(2) p Fn 911 2553 a(D) p Fo 995 2553 a(\() p Fn(H) p Fm 1114 2568 a(N) p Fo 1181 2553 a(\).) p 1324 2553 a(W) p 1416 2553 a(e) p 1503 2553 a(ha) m(v) m(e) p Fn 1740 2553 a(\021) p Fm 1788 2568 a(j) p Fn 1825 2553 a(u) p Fk 1927 2553 a(2) p Fn 2041 2553 a(D) p Fo 2125 2553 a(\() p Fn(L) p Fg 2229 2517 a(\003) p Fm 2229 2578 a(N) p Fo 2296 2553 a(\)) p 2379 2553 a(b) m(y) p 2526 2553 a(Lemma) p 2885 2553 a(4.3.) p 3115 2553 a(By) p 3280 2553 a(\(8.2\),) p 3554 2553 a(w) m(e) p 3710 2553 a(ha) m(v) m(e) 0 2673 y(\004) p Fl 65 2688 a(1) p Fo 104 2673 a(\() p Fn(T) p Fm 199 2688 a(j) p Fb 232 2669 a(0) p Fn 258 2673 a(u) p Fo(\)) p Fk 380 2673 a(2) p Fo 474 2673 a(Ran) p Fn 665 2673 a(M) p Fl 759 2688 a(1) p Fo 831 2673 a(for) p Fn 980 2673 a(j) p Fg 1026 2637 a(0) p Fo 1077 2673 a(=) p 1181 2673 a(1) p Fn(;) p 1274 2673 a(:) p 1318 2673 a(:) p 1362 2673 a(:) p 1404 2673 a(;) p 1448 2673 a(N) p Fo 1536 2673 a(.) p 1607 2673 a(Then,) 819 2893 y(\004) p Fl 884 2908 a(1) p Fo 923 2893 a(\() p Fn(t) p Fg 996 2908 a(\000) p Fm(z) p Fe 1084 2927 a(j) p Fb 1112 2913 a(0) p Fo 1143 2893 a(\() p Fn(\021) p Fm 1229 2908 a(j) p Fn 1266 2893 a(u) p Fo(\)\)) p 1425 2893 a(=) p 1528 2893 a(\004) p Fl 1593 2908 a(1) p Fo 1633 2893 a(\() p Fn(T) p Fm 1728 2908 a(j) p Fb 1761 2889 a(0) p Fo 1787 2893 a(\() p Fn(\021) p Fm 1873 2908 a(j) p Fn 1910 2893 a(u) p Fo(\)\)) p 2069 2893 a(=) p Fn 2172 2893 a(\016) p Fm 2215 2908 a(j) p Fb 2248 2889 a(0) p Fm 2270 2908 a(j) p Fo 2307 2893 a(\004) p Fl 2372 2908 a(1) p Fo 2411 2893 a(\() p Fn(T) p Fm 2506 2908 a(j) p Fn 2543 2893 a(u) p Fo(\)) p Fk 2664 2893 a(2) p Fo 2758 2893 a(Ran) p Fn 2949 2893 a(M) p Fl 3043 2908 a(1) p Fo 0 3113 a(for) p Fn 151 3113 a(j) p Fg 197 3077 a(0) p Fo 250 3113 a(=) p 356 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3354 a(the) p 561 3354 a(assertion) p 969 3354 a(\(i\)) p 1104 3354 a(holds.) p 1398 3354 a(The) p 1598 3354 a(assertion) p 2006 3354 a(\(ii\)) p 2168 3354 a(can) p 2347 3354 a(b) s(e) p 2480 3354 a(pro) m(v) m(ed) p 2797 3354 a(similarly) p 3159 3354 a(.) p 3226 3286 64 4 v 3226 3379 4 93 v 3286 3379 V 3226 3382 64 4 v Fq 146 3582 a(Lemma) p 543 3582 a(8.3) p Ff 735 3582 a(L) p 791 3582 a(et) p Fn 915 3582 a(U) p Ff 1039 3582 a(b) p 1079 3582 a(e) p 1170 3582 a(an) p 1321 3582 a(simply) p 1639 3582 a(c) p 1679 3582 a(onne) p 1879 3582 a(cte) p 1996 3582 a(d) p 2092 3582 a(op) p 2187 3582 a(en) p 2333 3582 a(set) p 2497 3582 a(in) p Fj 2628 3582 a(R) p Fl 2715 3540 a(2) p Fk 2786 3582 a(n) p Fn 2866 3582 a(S) p Fm 2926 3597 a(N) p Ff 2993 3582 a(,) p Fn 3073 3582 a(m) p Fk 3208 3582 a(2) p Fj 3324 3582 a(R) p Fl 3411 3540 a(2) p Ff 3497 3582 a(and) p Fn 3698 3582 a(V) p Fo 3826 3582 a(=) p Fn 0 3703 a(U) p Fk 98 3703 a(\000) p Fn 197 3703 a(m) p Fo 310 3703 a(=) p Fk 414 3703 a(f) p 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2318 3823 a(t) p Fm 2353 3838 a(m) p Ff 2454 3823 a(b) p 2494 3823 a(e) p 2573 3823 a(the) p 2734 3823 a(inverse) p 3068 3823 a(op) p 3163 3823 a(er) p 3244 3823 a(ator.) p 3492 3823 a(Then,) p 3775 3823 a(the) 0 3944 y(fol) p 110 3944 a(lowing) p 413 3944 a(holds.) 146 4064 y(\(i\)) p 291 4064 a(F) p 348 4064 a(or) p 473 4064 a(any) p Fn 659 4064 a(u) p Fk 743 4064 a(2) p Fn 837 4064 a(D) p Fo 921 4064 a(\() p Fn(H) p Fm 1040 4079 a(N) p Fo 1106 4064 a(\)) p Ff(,) p 1209 4064 a(we) p 1354 4064 a(have) p Fn 1578 4064 a(\021) p Fm 1626 4079 a(m) p Fn 1693 4064 a(u) p Fk 1776 4064 a(2) p Fn 1870 4064 a(D) p Fo 1954 4064 a(\() p Fn(H) p Fm 2073 4079 a(N) p Fo 2139 4064 a(\)) p Ff(,) p Fn 2242 4064 a(t) p Fg 2277 4079 a(\000) p Fm(m) p Fo 2399 4064 a(\() p Fn(\021) p Fm 2485 4079 a(m) p Fn 2552 4064 a(u) p Fo(\)) p Fk 2673 4064 a(2) p Fn 2767 4064 a(D) p Fo 2851 4064 a(\() p Fn(H) p Fl 2970 4079 a(0) p Fo 3009 4064 a(\)) p Ff 3081 4064 a(and) p Fn 1362 4284 a(H) p Fm 1443 4299 a(N) p Fo 1510 4284 a(\() p Fn(\021) p Fm 1596 4299 a(m) p Fn 1663 4284 a(u) p Fo(\)) p 1784 4284 a(=) p Fn 1888 4284 a(t) p Fm 1923 4299 a(m) p Fn 1990 4284 a(H) p Fl 2071 4299 a(0) p Fn 2110 4284 a(t) p Fg 2145 4299 a(\000) p Fm(m) p Fo 2266 4284 a(\() p Fn(\021) p Fm 2352 4299 a(m) p Fn 2419 4284 a(u) p Fo(\)) p Fn(:) p Fo 3701 4284 a(\(8.5\)) p Ff 146 4504 a(\(ii\)) p 321 4504 a(F) p 378 4504 a(or) p 503 4504 a(any) p Fn 689 4504 a(v) p Fk 768 4504 a(2) p Fn 862 4504 a(D) p Fo 946 4504 a(\() p Fn(H) p Fl 1065 4519 a(0) p Fo 1104 4504 a(\)) p Ff(,) p 1206 4504 a(we) p 1351 4504 a(have) p Fn 1575 4504 a(\021) t(v) p Fk 1705 4504 a(2) p Fn 1799 4504 a(D) p Fo 1883 4504 a(\() p Fn(H) p Fl 2002 4519 a(0) p Fo 2041 4504 a(\)) p Ff(,) p Fn 2144 4504 a(t) p Fm 2179 4519 a(m) p Fo 2246 4504 a(\() p Fn(\021) t(v) p Fo 2387 4504 a(\)) p Fk 2452 4504 a(2) p Fn 2546 4504 a(D) p Fo 2630 4504 a(\() p Fn(H) p Fm 2749 4519 a(N) p Fo 2815 4504 a(\)) p Ff 2888 4504 a(and) p Fn 1430 4724 a(t) p Fg 1465 4739 a(\000) p Fm(m) p Fn 1587 4724 a(H) p Fm 1668 4739 a(N) p Fn 1735 4724 a(t) p Fm 1770 4739 a(m) p Fo 1837 4724 a(\() p Fn(\021) t(v) p Fo 1978 4724 a(\)) p 2043 4724 a(=) p Fn 2146 4724 a(H) p Fl 2227 4739 a(0) p Fo 2266 4724 a(\() p Fn(\021) t(v) p Fo 2407 4724 a(\)) p Fn(:) p Fo 3701 4724 a(\(8.6\)) 1902 5214 y(46) p eop %%Page: 47 47 47 46 bop Ff 146 631 a(Pr) p 248 631 a(o) p 293 631 a(of.) p Fo 433 631 a(T) p 495 631 a(ak) m(e) p Fn 667 631 a(u) p Fk 750 631 a(2) p Fn 844 631 a(D) p Fo 928 631 a(\() p Fn(H) p Fm 1047 646 a(N) p Fo 1114 631 a(\).) p 1221 631 a(Then) p Fn 1474 631 a(\021) p Fm 1522 646 a(m) p Fn 1588 631 a(u) p Fk 1671 631 a(2) p Fn 1766 631 a(D) p Fo 1850 631 a(\() p 1888 553 134 4 v Fn(L) p Fm 1954 646 a(N) p Fo 2021 631 a(\)) p 2089 631 a(b) m(y) p 2222 631 a(\(ii\)) p 2382 631 a(of) p 2491 631 a(Prop) s(osition) p 3014 631 a(4.2.) p 3208 631 a(Since) p Fn 3460 631 a(H) p Fm 3541 646 a(N) p Fk 3636 631 a(\033) p 3741 553 V Fn 3741 631 a(L) p Fm 3807 646 a(N) p Fo 3875 631 a(,) 0 751 y(w) m(e) p 144 751 a(ha) m(v) m(e) p Fn 369 751 a(\021) p Fm 417 766 a(m) p Fn 483 751 a(u) p Fk 566 751 a(2) p Fn 661 751 a(D) p Fo 745 751 a(\() p Fn(H) p Fm 864 766 a(N) p Fo 930 751 a(\).) p 1039 751 a(Moreo) m(v) m(er,) p Fk 1170 931 a(L) p Fl 1239 946 a(0) p Fo 1278 931 a(\() p Fn(t) p Fg 1351 946 a(\000) p Fm(m) p Fn 1473 931 a(\021) p Fm 1521 946 a(m) p Fn 1587 931 a(u) p Fo(\)) p 1708 931 a(=) p Fn 1812 931 a(t) p Fg 1847 946 a(\000) p Fm(m) p Fk 1969 931 a(L) p Fm 2038 946 a(N) p Fn 2105 931 a(\021) p Fm 2153 946 a(m) p Fn 2219 931 a(u) p Fk 2303 931 a(2) p Fn 2397 931 a(L) p Fl 2463 890 a(2) p Fo 2502 931 a(\() p Fj(R) p Fl 2628 889 a(2) p Fo 2667 931 a(\)) p Fn(:) p Fo 0 1111 a(This) p 222 1111 a(implies) p Fn 552 1111 a(t) p Fg 587 1126 a(\000) p Fm(m) p Fn 709 1111 a(\021) p Fm 757 1126 a(m) p Fn 823 1111 a(u) p Fk 907 1111 a(2) p Fn 1001 1111 a(D) p Fo 1085 1111 a(\() p Fn(H) p Fl 1204 1126 a(0) p Fo 1242 1111 a(\)) p 1312 1111 a(and) p 1501 1111 a(\(8.5\),) p 1760 1111 a(since) p Fn 1998 1111 a(H) p Fl 2079 1126 a(0) p Fo 2150 1111 a(coincides) p 2561 1111 a(with) p 2783 1111 a(the) p 2950 1111 a(maximal) p 3347 1111 a(op) s(erator) p 3739 1111 a(\(see) 0 1232 y([Ik-Ka]\).) 146 1352 y(Th) m(us) p 393 1352 a(the) p 561 1352 a(assertion) p 969 1352 a(\(i\)) p 1104 1352 a(holds.) p 1398 1352 a(The) p 1598 1352 a(assertion) p 2006 1352 a(\(ii\)) p 2168 1352 a(can) p 2347 1352 a(b) s(e) p 2480 1352 a(pro) m(v) m(ed) p 2797 1352 a(similarly) p 3159 1352 a(.) p 3226 1284 64 4 v 3226 1377 4 93 v 3286 1377 V 3226 1380 64 4 v Ff 146 1535 a(Pr) p 248 1535 a(o) p 293 1535 a(of) p 408 1535 a(of) p 524 1535 a(The) p 684 1535 a(or) p 770 1535 a(em) p 929 1535 a(1.3.) p Fo 1166 1535 a(W) p 1258 1535 a(e) p 1335 1535 a(shall) p 1563 1535 a(in) m(tro) s(duce) p 1998 1535 a(the) p 2166 1535 a(notation) p 2557 1535 a(used) p 2780 1535 a(in) p 2895 1535 a([Co-Ne]) p 3250 1535 a(and) p 3440 1535 a(some) p 3685 1535 a(addi-) 0 1655 y(tional) p 276 1655 a(notations.) p 743 1655 a(F) p 799 1655 a(or) p Fn 918 1655 a(p) p Fo 994 1655 a(=) p 1098 1655 a(\(\() p Fn(p) p Fo(\)) p Fm 1261 1670 a(x) p Fn 1305 1655 a(;) p Fo 1349 1655 a(\() p Fn(p) p Fo(\)) p Fm 1474 1670 a(y) p Fo 1515 1655 a(\)) p Fk 1580 1655 a(2) p Fj 1674 1655 a(Z) p Fl 1758 1613 a(2) p Fo 1830 1655 a(and) p Fn 2020 1655 a(\016) p 2094 1655 a(>) p Fo 2198 1655 a(0,) p 2306 1655 a(put) p Fn 513 1887 a(K) p Fo 603 1887 a(\() p Fn(p;) p 734 1887 a(\016) p Fo 781 1887 a(\)) p 846 1887 a(=) p Fk 950 1887 a(f) p Fn(z) p Fo 1077 1887 a(=) p 1181 1887 a(\() p Fn(x;) p 1318 1887 a(y) p Fo 1370 1887 a(\)) p Fk 1435 1887 a(2) p Fj 1529 1887 a(R) p Fl 1616 1845 a(2) p Fo 1655 1887 a(;) p Fk 1732 1887 a(j) p Fn(x) p Fk 1837 1887 a(\000) p Fn 1982 1820 a(R) p 1946 1864 147 4 v Fo 1946 1956 a(100) 2103 1887 y(\() p Fn(p) p Fo(\)) p Fm 2228 1902 a(x) p Fk 2271 1887 a(j) p 2327 1887 a(\024) p Fn 2443 1820 a(\016) p 2442 1864 49 4 v Fo 2442 1956 a(2) p Fn 2501 1887 a(;) p Fk 2577 1887 a(j) p Fn(y) p Fk 2678 1887 a(\000) p Fn 2823 1820 a(R) p 2787 1864 147 4 v Fo 2787 1956 a(100) 2944 1887 y(\() p Fn(p) p Fo(\)) p Fm 3069 1902 a(y) p Fk 3110 1887 a(j) p 3165 1887 a(\024) p Fn 3281 1820 a(\016) p 3280 1864 49 4 v Fo 3280 1956 a(2) p Fk 3339 1887 a(g) p Fo 0 2108 a(and) p 190 2108 a(let) p Fn 331 2108 a(m) p Fo(\() p Fn(p) p Fo(\)) p 568 2108 a(=) p Fm 708 2069 a(R) p 682 2085 106 4 v Fl 682 2142 a(100) p Fn 798 2108 a(p) p Fo(.) p 917 2108 a(Since) p Fi 1615 2159 a([) p Fm 1577 2348 a(p) p Fg(2) p Fa(Z) p Fd 1710 2330 a(2) p Fn 1761 2242 a(K) p Fo 1851 2242 a(\() p Fn(p;) p 1982 2242 a(\016) p Fo 2029 2242 a(\)) p 2094 2242 a(=) p Fj 2198 2242 a(R) p Fl 2285 2200 a(2) p Fo 0 2500 a(for) p Fn 149 2500 a(\016) p Fk 224 2500 a(\025) p Fm 365 2460 a(R) p 339 2476 V Fl 339 2534 a(100) p Fo 455 2500 a(,) p 514 2500 a(w) m(e) p 658 2500 a(can) p 837 2500 a(tak) m(e) p Fn 1048 2500 a(p) p Fm 1097 2515 a(j) p Fk 1162 2500 a(2) p Fj 1256 2500 a(Z) p Fl 1340 2458 a(2) p Fo 1412 2500 a(suc) m(h) p 1632 2500 a(that) p Fn 1843 2500 a(z) p Fm 1888 2515 a(j) p Fk 1953 2500 a(2) p Fn 2047 2500 a(K) p Fo 2137 2500 a(\() p Fn(p) p Fm 2224 2515 a(j) p Fn 2261 2500 a(;) p Fm 2340 2460 a(R) p 2314 2476 V Fl 2314 2534 a(100) p Fo 2430 2500 a(\)) p 2501 2500 a(for) p 2650 2500 a(eac) m(h) p Fn 2869 2500 a(j) p Fo 2943 2500 a(=) p 3047 2500 a(1) p Fn(;) p 3140 2500 a(:) p 3184 2500 a(:) p 3228 2500 a(:) p 3270 2500 a(;) p 3314 2500 a(N) p Fo 3402 2500 a(.) p 3473 2500 a(F) p 3529 2500 a(or) p Fn 3648 2500 a(\016) p 3722 2500 a(>) p Fo 3826 2500 a(0,) 0 2620 y(put) p Fk 849 2800 a(K) p Fm 925 2815 a(j) p Fo 962 2800 a(\() p Fn(\016) p Fo 1047 2800 a(\)) p 1112 2800 a(:) p 1222 2800 a(=) p Fi 1545 2717 a([) p Fm 1381 2905 a(\014) s(;\015) p Fg 1484 2905 a(2f\000) p Fl(1) p Fm(;) p Fl(0) p Fm(;) p Fl(1) p Fg(g) p Fn 1818 2800 a(K) p Fo 1925 2800 a(\(\(\() p Fn(p) p Fm 2088 2815 a(j) p Fo 2124 2800 a(\)) p Fm 2162 2815 a(x) p Fo 2228 2800 a(+) p Fn 2326 2800 a(\014) p 2387 2800 a(;) p Fo 2431 2800 a(\() p Fn(p) p Fm 2518 2815 a(j) p Fo 2554 2800 a(\)) p Fm 2592 2815 a(y) p Fo 2655 2800 a(+) p Fn 2753 2800 a(\015) p Fo 2809 2800 a(\)) p Fn 2864 2800 a(;) p 2908 2800 a(\016) p Fo 2955 2800 a(\)) p Fn 3009 2800 a(;) p Fk 1000 3125 a(F) p Fm 1072 3140 a(N) p Fo 1222 3125 a(=) p Fm 1411 3017 a(N) p Fi 1396 3042 a([) p Fm 1381 3224 a(j) p Fl 1414 3224 a(=1) p Fk 1520 3125 a(K) p Fm 1596 3140 a(j) p Fo 1633 3125 a(\() p Fn 1716 3057 a(R) p 1681 3102 147 4 v Fo 1681 3193 a(100) 1837 3125 y(\)) 0 3388 y(and) p 190 3388 a(let) p Fn 331 3388 a(m) p Fm 416 3403 a(j) p Fo 480 3388 a(=) p Fn 584 3388 a(m) p Fo(\() p Fn(p) p Fm 756 3403 a(j) p Fo 792 3388 a(\).) p 901 3388 a(Let) 907 3619 y(\000) p Fl 968 3634 a(0) p Fo 1091 3619 a(=) p Fk 1250 3619 a(f) p Fn(m) p Fo(\() p Fn(p) p Fo(\)) p Fk 1537 3619 a(2) p Fj 1631 3619 a(R) p Fl 1719 3577 a(2) p Fo 1758 3619 a(;) p Fn 1834 3619 a(p) p Fk 1911 3619 a(2) p Fj 2005 3619 a(Z) p Fl 2089 3577 a(2) p Fn 2128 3619 a(;) p 2205 3619 a(K) p Fo 2295 3619 a(\() p Fn(p;) 2471 3552 y(R) p 2435 3596 V Fo 2435 3688 a(100) 2592 3619 y(\)) p Fk 2657 3619 a(6\032) p 2762 3619 a(F) p Fm 2834 3634 a(N) p Fk 2901 3619 a(g) p Fn(;) p Fo 907 3795 a(\000) p Fl 968 3810 a(1) p Fo 1091 3795 a(=) p Fk 1250 3795 a(f) p Fn(m) p Fo(\() p Fn(p) p Fm 1472 3810 a(j) p Fo 1508 3795 a(\);) p Fn 1622 3795 a(j) p Fo 1696 3795 a(=) p 1799 3795 a(1) p Fn(;) p 1892 3795 a(:) p 1936 3795 a(:) p 1980 3795 a(:) p 2023 3795 a(;) p 2067 3795 a(N) p Fk 2155 3795 a(g) p Fn(;) p Fo 947 3941 a(\000) p 1091 3941 a(=) p 1250 3941 a(\000) p Fl 1311 3956 a(0) p Fk 1372 3941 a([) p Fo 1461 3941 a(\000) p Fl 1522 3956 a(1) p Fn 1561 3941 a(:) p Ff 146 4123 a(Pr) p 248 4123 a(o) p 293 4123 a(of) p 412 4123 a(of) p 530 4123 a(\(i\)) p 679 4123 a(of) p 797 4123 a(The) p 957 4123 a(or) p 1043 4123 a(em) p 1206 4123 a(1.3) p 1374 4123 a(.) p Fo 1498 4123 a(Let) p Fn 1677 4123 a(I) p Fo 1763 4123 a(=) p 1873 4123 a([) p Fn(c;) p 1986 4123 a(d) p Fo(]) p 2101 4123 a(b) s(e) p 2238 4123 a(an) p 2378 4123 a(in) m(terv) p 2622 4123 a(al) p 2734 4123 a(on) p Fj 2874 4123 a(R) p Fo 2998 4123 a(satisfying) p Fn 3437 4123 a(I) p Fk 3513 4123 a(\\) p 3605 4123 a(f) p Fo(\(2) p Fn(n) p Fk 3825 4123 a(\000) p Fo 0 4244 a(1\)) p Fn(B) p Fo 166 4244 a(;) p Fn 210 4244 a(n) p Fo 296 4244 a(=) p 401 4244 a(1) p Fn(;) p Fo 494 4244 a(2) p Fn(;) p 587 4244 a(:) p 631 4244 a(:) p 675 4244 a(:) p Fk 700 4244 a(g) p Fo 779 4244 a(=) p Fk 883 4244 a(;) p Fo 966 4244 a(and) p Fn 1157 4244 a(c;) p 1243 4244 a(d) p 1333 4244 a(=) p Fk 1322 4244 a(2) p Fn 1417 4244 a(\033) p Fo 1476 4244 a(\() p Fn(H) p Fl 1595 4259 a(1) p Fo 1634 4244 a(\).) p 1744 4244 a(T) p 1806 4244 a(ak) m(e) p Fn 1980 4244 a(E) p Fk 2087 4244 a(2) p Fn 2182 4244 a(\033) p Fo 2241 4244 a(\() p Fn(H) p Fm 2360 4259 a(N) p Fo 2427 4244 a(\)) p Fk 2488 4244 a(\\) p Fn 2577 4244 a(I) p Fo 2628 4244 a(.) p 2700 4244 a(F) p 2756 4244 a(or) p 2875 4244 a(an) m(y) p Fn 3060 4244 a(\017) p 3128 4244 a(>) p Fo 3232 4244 a(0,) p 3342 4244 a(there) p 3591 4244 a(is) p 3690 4244 a(some) p Fn 0 4364 a( ) p Fm 63 4379 a(\017) p Fk 124 4364 a(2) p Fn 218 4364 a(D) p Fo 302 4364 a(\() p Fn(H) p Fm 421 4379 a(N) p Fo 488 4364 a(\)) p 558 4364 a(suc) m(h) p 778 4364 a(that) p Fk 1121 4544 a(jj) p Fn( ) t(\017) p Fk(jj) p Fo 1366 4544 a(=) p 1470 4544 a(1) p Fn(;) p Fk 1595 4544 a(jj) p Fn(\030) p Fm 1694 4559 a(\017) p Fk 1725 4544 a(jj) p Fn 1808 4544 a(<) p 1912 4544 a(\017;) p 2027 4544 a(\030) p Fm 2070 4559 a(\017) p Fo 2130 4544 a(=) p 2234 4544 a(\() p Fn(H) p Fm 2353 4559 a(N) p Fk 2442 4544 a(\000) p Fn 2542 4544 a(E) p Fo 2620 4544 a(\)) p Fn( ) p Fm 2721 4559 a(\017) p Fn 2754 4544 a(:) p Fo 3701 4544 a(\(8.7\)) 0 4724 y(Remark) p 365 4724 a(that) p 575 4724 a(when) p Fn 827 4724 a(N) p Fo 946 4724 a(is) p 1043 4724 a(\014nite,) p Fn 1317 4724 a(\033) p Fo 1376 4724 a(\() p Fn(H) p Fm 1495 4739 a(N) p Fo 1563 4724 a(\)) p Fk 1619 4724 a(\\) p Fn 1703 4724 a(I) p Fo 1785 4724 a(is) p 1881 4724 a(a) p 1960 4724 a(\014nite) p 2208 4724 a(set) p 2358 4724 a(b) m(y) p 2491 4724 a(\(iii\)) p 2679 4724 a(of) p 2788 4724 a(Theorem) p 3198 4724 a(1.2.) p 3393 4724 a(In) p 3513 4724 a(this) p 3701 4724 a(case,) 0 4844 y(w) m(e) p 144 4844 a(tak) m(e) p Fn 355 4844 a( ) p Fm 418 4859 a(\017) p Fo 484 4844 a(as) p 603 4844 a(an) p 739 4844 a(eigenfunction) p 1338 4844 a(corresp) s(onding) p 1964 4844 a(to) p 2083 4844 a(the) p 2251 4844 a(eigen) m(v) p 2511 4844 a(alue) p Fn 2717 4844 a(E) p Fo 2828 4844 a(and) p Fn 3018 4844 a(\030) p Fm 3061 4859 a(\017) p Fo 3126 4844 a(as) p 3245 4844 a(0.) 146 4965 y(W) p 238 4965 a(e) p 314 4965 a(shall) p 542 4965 a(prepare) p 895 4965 a(three) p 1144 4965 a(lemmas.) 1902 5214 y(47) p eop %%Page: 48 48 48 47 bop Fq 146 631 a(Lemma) p 543 631 a(8.4) p Ff 735 631 a(L) p 791 631 a(et) p Fn 904 631 a(\021) p Fk 983 631 a(2) p Fn 1077 631 a(C) p Fg 1154 595 a(1) p Fo 1229 631 a(\() p Fj(R) p Fl 1354 589 a(2) p Fo 1393 631 a(\)) p Ff(.) p 1506 631 a(Supp) p 1708 631 a(ose) p Fo 1877 631 a(supp) p Fk 2095 631 a(jr) p Fn(\021) p Fk 2258 631 a(j) p Ff 2320 631 a(is) p 2424 631 a(a) p 2509 631 a(c) p 2549 631 a(omp) p 2724 631 a(act) p 2885 631 a(set) p 3037 631 a(in) p Fj 3157 631 a(R) p Fl 3244 589 a(2) p Fk 3306 631 a(n) p Fn 3378 631 a(S) p Fm 3438 646 a(N) p Ff 3540 631 a(and) p Fo 1042 823 a(sup) p Fk 1206 823 a(jr) p Fn(\021) p Fk 1369 823 a(j) p Fl 1397 782 a(2) p Fo 1457 823 a(+) p 1555 823 a(sup) p Fk 1719 823 a(jr) p Fn(@) p Fm 1881 838 a(x) p Fn 1925 823 a(\021) p Fk 1977 823 a(j) p Fl 2005 782 a(2) p Fo 2066 823 a(+) p 2164 823 a(sup) p Fk 2327 823 a(jr) p Fn(@) p Fm 2489 838 a(y) p Fn 2531 823 a(\021) p Fk 2583 823 a(j) p Fl 2611 782 a(2) p Fk 2678 823 a(\024) p Fn 2783 823 a(C) p Ff 0 1016 a(for) p 162 1016 a(some) p 418 1016 a(c) p 458 1016 a(onstant) p Fn 813 1016 a(C) p Ff 890 1016 a(.) p 985 1016 a(Then,) p Fn 1277 1016 a(\021) t( ) p Fm 1392 1031 a(\017) p Fk 1465 1016 a(2) p Fn 1571 1016 a(D) p Fo 1655 1016 a(\() p Fn(H) p Fm 1774 1031 a(N) p Fo 1841 1016 a(\)) p Ff 1921 1016 a(and) p 2116 1016 a(ther) p 2279 1016 a(e) p 2366 1016 a(exists) p 2639 1016 a(a) p 2730 1016 a(c) p 2770 1016 a(onstant) p Fn 3126 1016 a(C) p Fl 3196 1031 a(5) p Fn 3275 1016 a(>) p Fo 3391 1016 a(0) p Ff 3481 1016 a(dep) p 3621 1016 a(endent) 0 1136 y(only) p 212 1136 a(on) p Fn 351 1136 a(E) p Ff 429 1136 a(,) p Fn 494 1136 a(C) p Ff 606 1136 a(such) p 827 1136 a(that) p Fk 848 1353 a(jj) p Fo([) p Fn(H) p Fm 1012 1368 a(N) p Fn 1079 1353 a(;) p 1123 1353 a(\021) p Fo 1175 1353 a(]) p Fn( ) p Fm 1265 1368 a(\017) p Fk 1297 1353 a(jj) p Fl 1353 1312 a(2) p Fk 1420 1353 a(\024) p Fn 1525 1353 a(C) p Fl 1595 1368 a(5) p Fi 1651 1236 a(Z) p Fl 1697 1424 a(supp) p Fg 1854 1424 a(jr) p Fm(\021) p Fg 1970 1424 a(j) p Fn 2011 1353 a(dxdy) p Fo 2220 1353 a(\() p Fk(j) p Fn( ) p Fm 2349 1368 a(\017) p Fo 2380 1353 a(\() p Fn(z) p Fo 2467 1353 a(\)) p Fk(j) p Fl 2533 1312 a(2) p Fo 2595 1353 a(+) p Fk 2693 1353 a(j) p Fn(\030) p Fm 2764 1368 a(\017) p Fo 2796 1353 a(\() p Fn(z) p Fo 2883 1353 a(\)) p Fk(j) p Fl 2949 1312 a(2) p Fo 2989 1353 a(\)) p Fn(:) p Fo 3701 1353 a(\(8.8\)) p Ff 146 1600 a(Pr) p 248 1600 a(o) p 293 1600 a(of.) p Fo 439 1600 a(First) p 680 1600 a(w) m(e) p 827 1600 a(sho) m(w) p Fn 1072 1600 a(\021) t( ) p Fm 1187 1615 a(\017) p Fk 1254 1600 a(2) p Fn 1354 1600 a(D) p Fo 1438 1600 a(\() p Fn(H) p Fm 1557 1615 a(N) p Fo 1623 1600 a(\).) p 1742 1600 a(Let) p Fn 1920 1600 a(\037) p Fm 1981 1615 a(j) p Fo 2054 1600 a(b) s(e) p 2190 1600 a(the) p 2362 1600 a(function) p 2747 1600 a(giv) m(en) p 3005 1600 a(in) p 3123 1600 a(Lemma) p 3474 1600 a(4.4.) p 3680 1600 a(Then) p Fn 0 1721 a(\037) p Fm 61 1736 a(j) p Fn 98 1721 a(\021) t( ) p Fm 213 1736 a(\017) p Fk 273 1721 a(2) p Fn 367 1721 a(D) p Fo 451 1721 a(\() p Fn(H) p Fm 570 1736 a(N) p Fo 637 1721 a(\)) p 707 1721 a(b) m(y) p 843 1721 a(Lemma) p 1191 1721 a(8.2.) p 1386 1721 a(Th) m(us) p Fn 1321 1990 a(S) p 1387 1990 a(T) p 1458 1990 a(\021) t( ) p Fm 1573 2005 a(\017) p Fo 1633 1990 a(=) p Fm 1766 1882 a(N) p Fi 1738 1907 a(X) p Fm 1737 2089 a(j) p Fl 1770 2089 a(=1) p Fn 1876 1990 a(\037) p Fl 1937 1949 a(2) p Fm 1937 2015 a(j) p Fn 1976 1990 a(\021) t( ) p Fm 2091 2005 a(\017) p Fk 2152 1990 a(2) p Fn 2246 1990 a(D) p Fo 2330 1990 a(\() p Fn(H) p Fm 2449 2005 a(N) p Fo 2516 1990 a(\)) p Fn(;) p Fo 0 2276 a(where) p Fn 284 2276 a(S) p Fo 350 2276 a(,) p Fn 412 2276 a(T) p Fo 517 2276 a(are) p 682 2276 a(the) p 852 2276 a(maps) p 1110 2276 a(de\014ned) p 1448 2276 a(in) p 1564 2276 a(Lemma) p 1914 2276 a(4.4.) p 2116 2276 a(Moreo) m(v) m(er,) p 2573 2276 a(w) m(e) p 2719 2276 a(ha) m(v) m(e) p 2946 2276 a(\(1) p Fk 3056 2276 a(\000) p Fn 3158 2276 a(S) p 3224 2276 a(T) p Fo 3295 2276 a(\)) p Fn(\021) t( ) p Fm 3448 2291 a(\017) p Fk 3511 2276 a(2) p Fn 3609 2276 a(D) p Fo 3693 2276 a(\() p 3731 2198 134 4 v Fn(L) p Fm 3797 2291 a(N) p Fo 3864 2276 a(\)) 0 2396 y(b) m(y) p 135 2396 a(the) p 303 2396 a(same) p 547 2396 a(pro) s(of) p 801 2396 a(as) p 921 2396 a(that) p 1132 2396 a(of) p 1243 2396 a(\(4.18\).) p 1562 2396 a(Since) p 1817 2318 V Fn 1817 2396 a(L) p Fm 1883 2411 a(N) p Fk 1978 2396 a(\032) p Fn 2083 2396 a(H) p Fm 2164 2411 a(N) p Fo 2231 2396 a(,) p 2290 2396 a(w) m(e) p 2434 2396 a(ha) m(v) m(e) p 2658 2396 a(\(1) p Fk 2766 2396 a(\000) p Fn 2865 2396 a(S) p 2931 2396 a(T) p Fo 3002 2396 a(\)) p Fn(\021) t( ) p Fm 3155 2411 a(\017) p Fk 3215 2396 a(2) p Fn 3309 2396 a(D) p Fo 3393 2396 a(\() p Fn(H) p Fm 3512 2411 a(N) p Fo 3579 2396 a(\).) p 3687 2396 a(Th) m(us) p Fn 0 2516 a(\021) t( ) p Fm 115 2531 a(\017) p Fo 175 2516 a(=) p 279 2516 a(\(1) p Fk 388 2516 a(\000) p Fn 487 2516 a(S) p 553 2516 a(T) p Fo 624 2516 a(\)) p Fn(\021) t( ) p Fm 777 2531 a(\017) p Fo 832 2516 a(+) p Fn 930 2516 a(S) p 996 2516 a(T) p 1067 2516 a(\021) t( ) p Fm 1182 2531 a(\017) p Fk 1242 2516 a(2) p Fn 1336 2516 a(D) p Fo 1420 2516 a(\() p Fn(H) p Fm 1539 2531 a(N) p Fo 1606 2516 a(\).) 146 2637 y(By) p 299 2637 a(\(4.5\),) p 560 2637 a(\(8.7\)) p 793 2637 a(and) p 982 2637 a(the) p 1150 2637 a(Sc) m(h) m(w) m(arz) p 1527 2637 a(inequalit) m(y) p 1940 2637 a(,) p 1998 2637 a(w) m(e) p 2142 2637 a(ha) m(v) m(e) p Fk 658 2859 a(jj) p Fo([) p Fn(H) p Fm 822 2874 a(N) p Fn 889 2859 a(;) p 933 2859 a(\021) p Fo 985 2859 a(]) p Fn( ) p Fm 1075 2874 a(\017) p Fk 1107 2859 a(jj) p Fl 1163 2818 a(2) p Fk 1285 2859 a(\024) p Fo 1446 2859 a(10) p Fn(C) p Fi 1637 2742 a(Z) p Fl 1683 2931 a(supp) p Fg 1840 2931 a(jr) p Fm(\021) p Fg 1956 2931 a(j) p Fo 1980 2859 a(\() p Fk(j) p Fn(H) p Fm 2127 2874 a(N) p Fn 2194 2859 a( ) p Fm 2257 2874 a(\017) p Fk 2290 2859 a(j) p Fl 2318 2818 a(2) p Fo 2379 2859 a(+) p Fk 2477 2859 a(j) p Fn( ) p Fm 2568 2874 a(\017) p Fk 2601 2859 a(j) p Fl 2629 2818 a(2) p Fo 2668 2859 a(\)) p Fn(dxdy) p Fk 1285 3098 a(\024) p Fo 1446 3098 a(10) p Fn(C) p Fi 1637 2980 a(Z) p Fl 1683 3169 a(supp) p Fg 1840 3169 a(jr) p Fm(\021) p Fg 1956 3169 a(j) p Fk 1980 3098 a(f) p Fo(\(2) p Fn(E) p Fl 2195 3056 a(2) p Fo 2256 3098 a(+) p 2354 3098 a(1\)) p Fk(j) p Fn( ) p Fm 2532 3113 a(\017) p Fk 2565 3098 a(j) p Fl 2593 3056 a(2) p Fo 2654 3098 a(+) p 2752 3098 a(2) p Fk(j) p Fn(\030) p Fm 2872 3113 a(\017) p Fk 2904 3098 a(j) p Fl 2932 3056 a(2) p Fo 2971 3098 a(\)) p Fn(dxdy) t(:) p Fo 0 3345 a(Th) m(us) p 247 3345 a(the) p 415 3345 a(assertion) p 823 3345 a(holds) p 1078 3345 a(with) p Fn 1300 3345 a(C) p Fl 1370 3360 a(5) p Fo 1437 3345 a(=) p 1541 3345 a(20\() p Fn(E) p Fl 1755 3309 a(2) p Fo 1816 3345 a(+) p 1914 3345 a(1\)) p Fn(C) p Fo 2078 3345 a(.) p 2148 3277 64 4 v 2148 3370 4 93 v 2208 3370 V 2148 3373 64 4 v Fq 146 3541 a(Lemma) p 543 3541 a(8.5) p Ff 735 3541 a(L) p 791 3541 a(et) p Fn 907 3541 a(p) p Fk 989 3541 a(2) p Fo 1088 3541 a(\000) p Fl 1149 3556 a(0) p Ff 1188 3541 a(.) p 1272 3541 a(Then,) p 1559 3541 a(ther) p 1722 3541 a(e) p 1805 3541 a(exist) p 2035 3541 a(c) p 2075 3541 a(onstants) p Fn 2466 3541 a(C) p Fl 2536 3556 a(6) p Fn 2608 3541 a(>) p Fo 2717 3541 a(0) p Ff 2804 3541 a(and) p Fn 2996 3541 a(c) p 3071 3541 a(>) p Fo 3180 3541 a(0) p Ff 3266 3541 a(dep) p 3406 3541 a(endent) p 3725 3541 a(only) 0 3662 y(on) p Fn 139 3662 a(B) p Ff 218 3662 a(,) p Fn 283 3662 a(E) p Ff 396 3662 a(such) p 618 3662 a(that) p Fi 544 3761 a(Z) p Fm 590 3950 a(K) p Fl 655 3950 a(\() p Fm(p;) p Fe 770 3923 a(R) p 747 3935 92 4 v Fd 747 3976 a(100) p Fl 848 3950 a(\)) p Fn 897 3879 a(dxdy) p Fk 1106 3879 a(j) p Fn( ) p Fm 1197 3894 a(\017) p Fo 1228 3879 a(\() p Fn(z) p Fo 1315 3879 a(\)) p Fk(j) p Fl 1381 3838 a(2) p Fk 384 4159 a(\024) p Fn 544 4159 a(C) p Fl 614 4174 a(6) p Fi 670 4013 a( ) p Fn 736 4159 a(e) p Fg 781 4118 a(\000) p Fm(cR) p Fd 920 4094 a(2) p Fi 975 4042 a(Z) p 1022 4159 550 4 v Fm 1022 4230 a(K) p Fl 1087 4230 a(\() p Fm(p;) p Fe 1186 4203 a(R) p 1178 4215 61 4 v Fd 1178 4256 a(98) p Fl 1249 4230 a(\)) p Fg(n) p Fm(K) p Fl 1376 4230 a(\() p Fm(p;) p Fe 1476 4203 a(R) p 1469 4215 V Fd 1469 4256 a(99) p Fl 1539 4230 a(\)) p Fn 1588 4159 a(dxdy) p Fk 1797 4159 a(j) p Fn( ) p Fm 1888 4174 a(\017) p Fo 1919 4159 a(\() p Fn(z) p Fo 2006 4159 a(\)) p Fk(j) p Fl 2072 4118 a(2) p Fo 2134 4159 a(+) p Fi 2232 4042 a(Z) p Fm 2278 4230 a(K) p Fl 2343 4230 a(\() p Fm(p;) p Fe 2443 4203 a(R) p 2435 4215 V Fd 2435 4256 a(98) p Fl 2506 4230 a(\)) p Fn 2554 4159 a(dxdy) p Fk 2763 4159 a(j) p Fn(\030) p Fm 2834 4174 a(\017) p Fo 2865 4159 a(\() p Fn(z) p Fo 2952 4159 a(\)) p Fk(j) p Fl 3018 4118 a(2) p Fi 3058 4013 a(!) p Fn 3140 4159 a(;) p Fo 3219 4159 a(if) p Fn 3317 4159 a(R) p Fk 3420 4159 a(\025) p Fo 3525 4159 a(1) p Fn(:) p Fo 3701 4159 a(\(8.9\)) p Ff 146 4412 a(Pr) p 248 4412 a(o) p 293 4412 a(of.) p Fo 435 4412 a(T) p 497 4412 a(ak) m(e) p 671 4412 a(a) p 753 4412 a(smo) s(oth) p 1097 4412 a(function) p Fn 1479 4412 a(\021) p Fl 1527 4427 a(0) p Fo 1599 4412 a(satisfying) p 2034 4412 a(0) p Fk 2110 4412 a(\024) p Fn 2216 4412 a(\021) p Fl 2264 4427 a(0) p Fk 2331 4412 a(\024) p Fo 2436 4412 a(1) p 2517 4412 a(and) p Fn 1029 4661 a(\021) p Fl 1077 4676 a(0) p Fo 1117 4661 a(\() p Fn(z) p Fo 1204 4661 a(\)) p 1270 4661 a(=) p Fi 1374 4515 a(\() p Fo 1457 4601 a(1) p Fn 1604 4601 a(z) p Fk 1681 4601 a(2) p Fn 1775 4601 a(K) p Fo 1865 4601 a(\(0) p Fn(;) p Fm 2014 4562 a(R) p 2006 4578 71 4 v Fl 2006 4636 a(99) p Fo 2086 4601 a(\)) 1457 4722 y(0) p Fn 1604 4722 a(z) p 1693 4722 a(=) p Fk 1681 4722 a(2) p Fn 1775 4722 a(K) p Fo 1865 4722 a(\(0) p Fn(;) p Fm 2014 4682 a(R) p 2006 4698 V Fl 2006 4756 a(98) p Fo 2086 4722 a(\)) p Fn 2183 4661 a(;) p Fo 1029 4930 a(sup) p Fk 1193 4930 a(jr) p Fn(\021) p Fl 1352 4945 a(0) p Fk 1391 4930 a(j) p Fl 1419 4888 a(2) p Fo 1480 4930 a(+) p 1578 4930 a(sup) p Fk 1742 4930 a(jr) p Fn(@) p Fm 1904 4945 a(x) p Fn 1948 4930 a(\021) p Fl 1996 4945 a(0) p Fk 2035 4930 a(j) p Fl 2063 4888 a(2) p Fo 2125 4930 a(+) p 2223 4930 a(sup) p Fk 2386 4930 a(jr) p Fn(@) p Fm 2548 4945 a(y) p Fn 2590 4930 a(\021) p Fl 2638 4945 a(0) p Fk 2677 4930 a(j) p Fl 2705 4888 a(2) p Fk 2772 4930 a(\024) p Fn 2906 4862 a(C) p 2887 4906 115 4 v 2887 4998 a(R) p Fl 2962 4969 a(2) p Fn 3012 4930 a(;) p Fo 1902 5214 a(48) p eop %%Page: 49 49 49 48 bop Fo 0 631 a(where) p Fn 281 631 a(C) p Fo 390 631 a(is) p 487 631 a(a) p 568 631 a(p) s(ositiv) m(e) p 928 631 a(constan) m(t) p 1321 631 a(indep) s(enden) m(t) p 1873 631 a(of) p Fn 1983 631 a(R) p Fo 2058 631 a(.) p 2129 631 a(Put) p Fn 2319 631 a(\021) p Fm 2367 646 a(m) p Fo 2434 631 a(\() p Fn(z) p Fo 2521 631 a(\)) p 2587 631 a(=) p Fn 2691 631 a(\021) p Fl 2739 646 a(0) p Fo 2778 631 a(\() p Fn(z) p Fk 2887 631 a(\000) p Fn 2985 631 a(m) p Fo(\),) p 3167 631 a(where) p Fn 3448 631 a(m) p Fo 3561 631 a(=) p Fn 3665 631 a(m) p Fo(\() p Fn(p) p Fo(\),) p Fn 0 751 a(p) p Fk 102 751 a(2) p Fo 221 751 a(\000) p Fl 282 766 a(0) p Fo 321 751 a(.) p 436 751 a(Applying) p 873 751 a(Lemma) p 1236 751 a(8.3) p 1408 751 a(with) p Fn 1645 751 a(U) p Fo 1775 751 a(=) p Fn 1903 751 a(K) p Fo 1993 751 a(\() p Fn(p;) p Fm 2143 712 a(R) p 2134 728 71 4 v Fl 2134 786 a(98) p Fo 2215 751 a(\),) p 2331 751 a(w) m(e) p 2489 751 a(ha) m(v) m(e) p 2729 751 a(that) p Fn 2955 751 a(\021) p Fm 3003 766 a(m) p Fn 3069 751 a( ) p Fm 3132 766 a(\017) p Fk 3218 751 a(2) p Fn 3337 751 a(D) p Fo 3421 751 a(\() p Fn(H) p Fm 3540 766 a(N) p Fo 3607 751 a(\),) p 3723 751 a(that) p Fn 0 872 a(t) p Fg 35 887 a(\000) p Fm(m) p Fn 157 872 a(\021) p Fm 205 887 a(m) p Fn 271 872 a( ) p Fm 334 887 a(\017) p Fk 395 872 a(2) p Fn 489 872 a(D) p Fo 573 872 a(\() p Fn(H) p Fl 692 887 a(0) p Fo 731 872 a(\)) p 802 872 a(and) p 991 872 a(that) 597 1092 y([) p Fn(H) p Fm 705 1107 a(N) p Fn 772 1092 a(;) p 816 1092 a(\021) p Fm 864 1107 a(m) p Fo 931 1092 a(]) p Fn( ) p Fm 1021 1107 a(\017) p Fo 1076 1092 a(+) p Fn 1174 1092 a(\021) p Fm 1222 1107 a(m) p Fn 1289 1092 a(\030) p Fm 1332 1107 a(\017) p Fo 1392 1092 a(=) p 1495 1092 a(\() p Fn(H) p Fm 1614 1107 a(N) p Fk 1703 1092 a(\000) p Fn 1803 1092 a(E) p Fo 1881 1092 a(\)\() p Fn(\021) p Fm 2005 1107 a(m) p Fn 2072 1092 a( ) p Fm 2135 1107 a(\017) p Fo 2168 1092 a(\)) p 2233 1092 a(=) p Fn 2337 1092 a(t) p Fm 2372 1107 a(m) p Fo 2439 1092 a(\() p Fn(H) p Fl 2558 1107 a(0) p Fk 2619 1092 a(\000) p Fn 2719 1092 a(E) p Fo 2797 1092 a(\)) p Fn(t) p Fg 2870 1107 a(\000) p Fm(m) p Fo 2991 1092 a(\() p Fn(\021) p Fm 3077 1107 a(m) p Fn 3144 1092 a( ) p Fm 3207 1107 a(\017) p Fo 3240 1092 a(\)) p Fn(:) p Fo 3653 1092 a(\(8.10\)) 0 1312 y(Since) p 260 1312 a(the) p 433 1312 a(supp) s(ort) p 799 1312 a(of) p 915 1312 a(b) s(oth) p 1150 1312 a(sides) p 1389 1312 a(of) p 1505 1312 a(\(8.10\)) p 1792 1312 a(are) p 1960 1312 a(concluded) p 2420 1312 a(in) p Fn 2539 1312 a(K) p Fo 2629 1312 a(\() p Fn(p;) p Fm 2778 1273 a(R) p 2769 1289 V Fl 2769 1346 a(98) p Fo 2850 1312 a(\),) p 2954 1312 a(w) m(e) p 3102 1312 a(can) p 3286 1312 a(op) s(erate) p 3641 1312 a(\() p Fn(H) p Fl 3760 1327 a(0) p Fk 3825 1312 a(\000) p Fn 0 1432 a(E) p Fo 78 1432 a(\)) p Fg 116 1396 a(\000) p Fl(1) p Fn 210 1432 a(t) p Fg 245 1447 a(\000) p Fm(m) p Fo 400 1432 a(to) p 519 1432 a(b) s(oth) p 749 1432 a(sides) p 983 1432 a(of) p 1094 1432 a(\(8.10\)) p 1376 1432 a(and) p 1566 1432 a(obtain) 897 1652 y(\() p Fn(H) p Fl 1016 1667 a(0) p Fk 1077 1652 a(\000) p Fn 1177 1652 a(E) p Fo 1255 1652 a(\)) p Fg 1293 1611 a(\000) p Fl(1) p Fn 1387 1652 a(t) p Fg 1422 1667 a(\000) p Fm(m) p Fo 1561 1652 a(\() o([) p Fn(H) p Fm 1706 1667 a(N) p Fn 1774 1652 a(;) p 1818 1652 a(\021) p Fm 1866 1667 a(m) p Fo 1932 1652 a(]) p Fn( ) p Fm 2022 1667 a(\017) p Fo 2077 1652 a(+) p Fn 2175 1652 a(\021) p Fm 2223 1667 a(m) p Fn 2290 1652 a(\030) p Fm 2333 1667 a(\017) p Fo 2365 1652 a(\)) p 2431 1652 a(=) p Fn 2535 1652 a(t) p Fg 2570 1667 a(\000) p Fm(m) p Fo 2691 1652 a(\() p Fn(\021) p Fm 2777 1667 a(m) p Fn 2844 1652 a( ) p Fm 2907 1667 a(\017) p Fo 2940 1652 a(\)) p Fn(:) p Fo 3653 1652 a(\(8.11\)) 0 1872 y(Since) p 263 1872 a(the) p 438 1872 a(supp) s(ort) p 807 1872 a(of) p 926 1872 a(the) p 1102 1872 a(righ) m(t) p 1346 1872 a(hand) p 1598 1872 a(side) p 1801 1872 a(of) p 1920 1872 a(\(8.11\)) p 2210 1872 a(is) p 2316 1872 a(concluded) p 2779 1872 a(in) p Fn 2901 1872 a(K) p Fo 2991 1872 a(\(0) p Fn(;) p Fm 3157 1833 a(R) p 3131 1849 106 4 v Fl 3131 1907 a(100) p Fo 3247 1872 a(\),) p 3354 1872 a(so) p 3482 1872 a(is) p 3588 1872 a(the) p 3764 1872 a(left) 0 1993 y(hand) p 244 1993 a(side.) p 477 1993 a(Th) m(us) p 725 1993 a(w) m(e) p 868 1993 a(can) p 1047 1993 a(op) s(erate) p Fn 1397 1993 a(t) p Fm 1432 2008 a(m) p Fo 1531 1993 a(to) p 1650 1993 a(b) s(oth) p 1880 1993 a(sides) p 2115 1993 a(of) p 2226 1993 a(\(8.11\)) p 2507 1993 a(and) p 2697 1993 a(obtain) p Fn 962 2213 a(t) p Fm 997 2228 a(m) p Fo 1064 2213 a(\() p Fn(H) p Fl 1183 2228 a(0) p Fk 1245 2213 a(\000) p Fn 1344 2213 a(E) p Fo 1422 2213 a(\)) p Fg 1460 2171 a(\000) p Fl(1) p Fn 1554 2213 a(t) p Fg 1589 2228 a(\000) p Fm(m) p Fo 1728 2213 a(\([) p Fn(H) p Fm 1874 2228 a(N) p Fn 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2447 a(\(0) p Fn(;) p Fm 2666 2407 a(R) p 2640 2423 106 4 v Fl 2640 2481 a(100) p Fo 2756 2447 a(\)) p Fn(;) p 2838 2352 534 4 v 2838 2447 a(K) p Fo 2928 2447 a(\(0) p Fn(;) p Fm 3076 2407 a(R) p 3068 2423 71 4 v Fl 3068 2481 a(98) p Fk 3171 2447 a(n) p Fm 3261 2407 a(R) p 3252 2423 V Fl 3252 2481 a(99) p Fo 3333 2447 a(\)\)) p 3437 2447 a(=) p Fm 3611 2407 a(R) p 3550 2423 177 4 v Fl 3550 2481 a(19800) p Fo 3736 2447 a(,) p 3791 2447 a(w) m(e) 0 2567 y(ha) m(v) m(e) p 228 2567 a(b) m(y) p 366 2567 a([Co-Ne,) p 724 2567 a(Corollary) p 1157 2567 a(I) s(I.1]) p 1369 2567 a(that) p 1583 2567 a(there) p 1835 2567 a(exists) p 2107 2567 a(a) p 2191 2567 a(constan) m(t) p Fn 2587 2567 a(c) p 2661 2567 a(>) p Fo 2769 2567 a(0) p 2853 2567 a(dep) s(enden) m(t) p 3327 2567 a(only) p 3544 2567 a(on) p Fn 3682 2567 a(B) p Fo 3761 2567 a(,) p Fn 3824 2567 a(E) p Fo 0 2687 a(suc) m(h) p 220 2687 a(that) p Fi 501 2790 a(Z) p Fm 547 2979 a(K) p Fl 612 2979 a(\() p Fm(p;) p Fe 726 2952 a(R) p 704 2964 92 4 v Fd 704 3005 a(100) p Fl 805 2979 a(\)) p Fn 853 2907 a(dxdy) p Fk 1062 2907 a(j) p Fn(t) p Fm 1125 2922 a(m) p Fo 1191 2907 a(\() p Fn(H) p Fl 1310 2922 a(0) p Fk 1371 2907 a(\000) p Fn 1471 2907 a(E) p Fo 1549 2907 a(\)) p Fg 1587 2866 a(\000) p Fl(1) p Fn 1681 2907 a(t) p Fg 1716 2922 a(\000) p Fm(m) p Fo 1838 2907 a([) p Fn(H) p Fm 1946 2922 a(N) p Fn 2013 2907 a(;) p 2057 2907 a(\021) p Fm 2105 2922 a(m) p Fo 2171 2907 a(]) p Fn( ) p Fm 2261 2922 a(\017) p Fk 2294 2907 a(j) p Fl 2322 2866 a(2) p Fk 2389 2907 a(\024) p Fn 2494 2907 a(C) p 2571 2907 a(e) p Fg 2616 2866 a(\000) p Fm(cR) p Fd 2755 2843 a(2) p Fk 2794 2907 a(jj) p Fo([) p Fn(H) p Fm 2958 2922 a(N) p Fn 3025 2907 a(;) p 3069 2907 a(\021) p Fm 3117 2922 a(m) p Fo 3183 2907 a(]) p Fn( ) p Fm 3273 2922 a(\017) p Fk 3306 2907 a(jj) p Fl 3362 2866 a(2) p Fo 3653 2907 a(\(8.13\)) 0 3182 y(for) p 149 3182 a(ev) m(ery) p Fn 406 3182 a(R) p Fo 481 3182 a(.) p 552 3182 a(Th) m(us) p 799 3182 a(w) m(e) p 943 3182 a(ha) m(v) m(e) p Fi 647 3309 a(Z) p Fm 693 3498 a(K) p Fl 758 3498 a(\() p Fm(p;) p Fe 872 3471 a(R) p 849 3483 V Fd 849 3524 a(100) p Fl 951 3498 a(\)) p Fk 999 3426 a(j) p Fn( ) p Fm 1090 3441 a(\017) p Fk 1123 3426 a(j) p Fl 1151 3385 a(2) p Fn 1190 3426 a(dxdy) p Fo 1425 3426 a(=) p Fi 1529 3309 a(Z) p Fm 1575 3498 a(K) p Fl 1640 3498 a(\() p Fm(p;) p Fe 1754 3471 a(R) p 1732 3483 V Fd 1732 3524 a(100) p Fl 1833 3498 a(\)) p Fk 1881 3426 a(j) p Fn(\021) p Fm 1957 3441 a(m) p Fn 2024 3426 a( ) p Fm 2087 3441 a(\017) p Fk 2120 3426 a(j) p Fl 2148 3385 a(2) p Fn 2187 3426 a(dxdy) p Fk 486 3654 a(\024) p Fo 647 3654 a(2) p Fn(C) p 773 3654 a(e) p Fg 818 3613 a(\000) p Fm(cR) p Fd 957 3589 a(2) p Fk 995 3654 a(jj) p Fo([) p Fn(H) p Fm 1159 3669 a(N) p Fn 1226 3654 a(;) p 1270 3654 a(\021) p Fm 1318 3669 a(m) p Fo 1384 3654 a(]) p Fn( ) p Fm 1474 3669 a(\017) p Fk 1507 3654 a(jj) p Fl 1563 3613 a(2) p Fo 1624 3654 a(+) p 1722 3654 a(2) p Fk(jj) p Fn(t) p Fm 1862 3669 a(m) p Fo 1928 3654 a(\() p Fn(H) p Fl 2047 3669 a(0) p Fk 2109 3654 a(\000) p Fn 2208 3654 a(E) p Fo 2286 3654 a(\)) p Fg 2324 3613 a(\000) p Fl(1) p Fn 2419 3654 a(t) p Fg 2454 3669 a(\000) p Fm(m) p Fo 2575 3654 a(\() p Fn(\021) p Fm 2661 3669 a(m) p Fn 2728 3654 a(\030) p Fm 2771 3669 a(\017) p Fo 2803 3654 a(\)) p Fk(jj) p Fl 2897 3613 a(2) p Fk 486 3829 a(\024) p Fo 647 3829 a(2) p Fn(C) p 773 3829 a(C) p Fl 843 3844 a(5) p Fn 882 3829 a(e) p Fg 927 3788 a(\000) p Fm(cR) p Fd 1066 3765 a(2) p Fi 1121 3712 a(Z) p Fl 1167 3901 a(supp) p Fg 1324 3901 a(jr) p Fm(\021) p Fe 1438 3909 a(m) p Fg 1497 3901 a(j) p Fn 1537 3829 a(dxdy) p Fo 1746 3829 a(\() p Fk(j) p Fn( ) p Fm 1875 3844 a(\017) p Fk 1907 3829 a(j) p Fl 1935 3788 a(2) p Fo 1996 3829 a(+) p Fk 2094 3829 a(j) p Fn(\030) p Fm 2165 3844 a(\017) p Fk 2197 3829 a(j) p Fl 2225 3788 a(2) p Fo 2264 3829 a(\)) 647 4068 y(+2) p 789 4068 a(dist) o(\() p Fn(\033) p Fo 1043 4068 a(\() p Fn(H) p Fl 1162 4083 a(0) p Fo 1201 4068 a(\)) p Fn(;) p 1283 4068 a(E) p Fo 1361 4068 a(\)) p Fg 1399 4026 a(\000) p Fl(1) p Fi 1509 3950 a(Z) p Fl 1555 4139 a(supp) p Fm 1713 4139 a(\021) p Fe 1748 4147 a(m) p Fn 1827 4068 a(dxdy) p Fk 2036 4068 a(j) p Fn(\030) p Fm 2107 4083 a(\017) p Fk 2138 4068 a(j) p Fl 2166 4026 a(2) p Fk 486 4335 a(\024) p Fn 647 4335 a(C) p Fl 717 4350 a(6) p Fi 772 4189 a( ) p Fn 838 4335 a(e) p Fg 883 4294 a(\000) p Fm(cR) p Fd 1022 4270 a(2) p Fi 1078 4218 a(Z) p Fl 1124 4406 a(supp) p Fg 1281 4406 a(jr) p Fm(\021) p Fe 1395 4414 a(m) p Fg 1454 4406 a(j) p Fn 1494 4335 a(dxdy) p Fk 1703 4335 a(j) p Fn( ) p Fm 1794 4350 a(\017) p Fo 1826 4335 a(\() p Fn(z) p Fo 1913 4335 a(\)) p Fk(j) p Fl 1979 4294 a(2) p Fo 2040 4335 a(+) p Fi 2138 4218 a(Z) p Fl 2185 4406 a(supp) p Fm 2342 4406 a(\021) p Fe 2377 4414 a(m) p Fn 2457 4335 a(dxdy) p Fk 2666 4335 a(j) p Fn(\030) p Fm 2737 4350 a(\017) p Fo 2768 4335 a(\() p Fn(z) p Fo 2855 4335 a(\)) p Fk(j) p Fl 2921 4294 a(2) p Fi 2960 4189 a(!) p Fn 3043 4335 a(;) p Fo 3119 4335 a(if) p Fn 3215 4335 a(R) p Fk 3318 4335 a(\025) p Fo 3423 4335 a(1) p Fn(;) p Fo 0 4617 a(where) p Fn 275 4617 a(C) p Fl 345 4632 a(6) p Fo 410 4617 a(is) p 502 4617 a(a) p 576 4617 a(constan) m(t) p 963 4617 a(dep) s(enden) m(t) p 1428 4617 a(only) p 1635 4617 a(on) p Fn 1764 4617 a(B) p Fo 1843 4617 a(,) p Fn 1898 4617 a(E) p Fo 1976 4617 a(.) p 2044 4617 a(W) p 2136 4617 a(e) p 2205 4617 a(used) p 2421 4617 a(\(8.12\)) p 2697 4617 a(in) p 2804 4617 a(the) p 2965 4617 a(\014rst) p 3159 4617 a(inequalit) m(y) p 3572 4617 a(,) p 3626 4617 a(\(8.13\),) 0 4737 y(Lemma) p 341 4737 a(8.4) p 490 4737 a(and) p 672 4737 a(the) p 832 4737 a(equalit) m(y) p Fk 1196 4737 a(jj) p Fo(\() p Fn(H) p Fl 1371 4752 a(0) p Fk 1416 4737 a(\000) p Fn 1499 4737 a(E) p Fo 1577 4737 a(\)) p Fg 1615 4701 a(\000) p Fl(1) p Fk 1710 4737 a(jj) p Fo 1793 4737 a(=) p Fn 1897 4737 a(dist) p Fo(\() p Fn(\033) p Fo 2159 4737 a(\() p Fn(H) p Fl 2278 4752 a(0) p Fo 2317 4737 a(\)) p Fn(;) p 2399 4737 a(E) p Fo 2477 4737 a(\)) p Fg 2515 4701 a(\000) p Fl(1) p Fo 2634 4737 a(in) p 2740 4737 a(the) p 2900 4737 a(second) p 3208 4737 a(inequalit) m(y) p 3621 4737 a(.) p 3687 4737 a(Th) m(us) 0 4858 y(w) m(e) p 144 4858 a(ha) m(v) m(e) p 369 4858 a(the) p 537 4858 a(conclusion.) p 1046 4790 64 4 v 1046 4883 4 93 v 1107 4883 V 1046 4886 64 4 v 1902 5214 a(49) p eop %%Page: 50 50 50 49 bop Fq 146 631 a(Lemma) p 543 631 a(8.6) p Ff 735 631 a(Ther) p 936 631 a(e) p 1016 631 a(exists) p 1282 631 a(c) p 1322 631 a(onstants) p Fn 1711 631 a(C) p Fl 1781 646 a(7) p Fn 1847 631 a(>) p Fo 1951 631 a(0) p Ff 2035 631 a(dep) p 2175 631 a(endent) p 2490 631 a(only) p 2702 631 a(on) p Fn 2842 631 a(B) p Ff 2921 631 a(,) p Fn 2986 631 a(E) p Ff 3098 631 a(such) p 3320 631 a(that) p Fi 1138 745 a(Z) p Fg 1184 934 a(F) p Fe 1241 911 a(c) 1234 958 y(N) p Fn 1313 862 a(dxdy) p Fk 1522 862 a(j) p Fn( ) p Fm 1613 877 a(\017) p Fo 1645 862 a(\() p Fn(z) p Fo 1732 862 a(\)) p Fk(j) p Fl 1798 821 a(2) p Fk 1865 862 a(\024) p Fn 1970 862 a(C) p Fl 2040 877 a(7) p Fn 2080 862 a(e) p Fg 2125 821 a(\000) p Fm(cR) p Fd 2264 798 a(2) p Fn 2303 862 a(;) p Fo 2381 862 a(if) p Fn 2480 862 a(R) p Fk 2583 862 a(\025) p Fo 2688 862 a(1) p Fn(;) p Fo 3653 862 a(\(8.14\)) p Ff 0 1145 a(wher) p 196 1145 a(e) p Fn 275 1145 a(\017) p Fo 342 1145 a(=) p Fn 446 1145 a(e) p Fg 491 1108 a(\000) p Fe 557 1081 a(c) p 556 1093 31 4 v Fd 556 1134 a(2) p Fm 596 1108 a(R) p Fd 649 1085 a(2) p Ff 689 1145 a(,) p Fn 753 1145 a(c) p Ff 830 1145 a(is) p 935 1145 a(a) p 1020 1145 a(c) p 1060 1145 a(onstant) p 1408 1145 a(given) p 1662 1145 a(in) p 1782 1145 a(L) p 1838 1145 a(emma) p 2127 1145 a(8.5) p 2291 1145 a(and) p Fn 2480 1145 a( ) p Fm 2543 1160 a(\017) p Ff 2611 1145 a(satisfying) p 3045 1145 a(\(8.7\).) 146 1358 y(Pr) p 248 1358 a(o) p 293 1358 a(of.) p Fo 435 1358 a(Summing) p 869 1358 a(up) p 1010 1358 a(the) p 1178 1358 a(inequalit) m(y) p 1630 1358 a(\(8.9\)) p 1863 1358 a(for) p 2013 1358 a(all) p Fn 2148 1358 a(p) p Fk 2225 1358 a(2) p Fo 2319 1358 a(\000) p Fl 2380 1373 a(0) p Fo 2419 1358 a(,) p 2479 1358 a(w) m(e) p 2622 1358 a(ha) m(v) m(e) p Fi 525 1477 a(Z) p Fg 572 1666 a(F) p Fe 629 1643 a(c) 622 1691 y(N) p Fn 700 1595 a(dxdy) p Fk 909 1595 a(j) p Fn( ) p Fm 1000 1610 a(\017) p Fo 1032 1595 a(\() p Fn(z) p Fo 1119 1595 a(\)) p Fk(j) p Fl 1185 1553 a(2) p Fk 1253 1595 a(\024) p Fo 1358 1595 a(4) p Fn(C) p Fl 1477 1610 a(6) p Fo 1516 1595 a(\() p Fn(e) p Fg 1599 1553 a(\000) p Fm(cR) p Fd 1738 1530 a(2) p Fk 1777 1595 a(jj) p Fn( ) p Fm 1896 1610 a(\017) p Fk 1928 1595 a(jj) p Fl 1984 1553 a(2) p Fo 2045 1595 a(+) p Fk 2143 1595 a(jj) p Fn(\030) p Fm 2242 1610 a(\017) p Fk 2274 1595 a(jj) p Fl 2330 1553 a(2) p Fo 2368 1595 a(\)) p Fk 2434 1595 a(\024) p Fo 2539 1595 a(8) p Fn(C) p Fl 2658 1610 a(6) p Fn 2697 1595 a(e) p Fg 2742 1553 a(\000) p Fm(cR) p Fd 2881 1530 a(2) p Fn 2920 1595 a(;) p Fo 2996 1595 a(if) p Fn 3093 1595 a(R) p Fk 3195 1595 a(\025) p Fo 3301 1595 a(1) p Fn(:) p Fo 0 1870 a(In) p 126 1870 a(the) p 298 1870 a(\014rst) p 503 1870 a(inequalit) m(y) p 916 1870 a(,) p 980 1870 a(w) m(e) p 1128 1870 a(used) p 1355 1870 a(the) p 1527 1870 a(fact) p 1724 1870 a(an) m(y) p 1912 1870 a(p) s(oin) m(t) p Fn 2171 1870 a(z) p Fk 2255 1870 a(2) p Fj 2357 1870 a(R) p Fl 2444 1828 a(2) p Fo 2520 1870 a(is) p 2622 1870 a(con) m(tained) p 3068 1870 a(in) p Fn 3186 1870 a(K) p Fo 3276 1870 a(\() p Fn(p;) p Fm 3425 1831 a(R) p 3417 1847 71 4 v Fl 3417 1904 a(98) p Fo 3497 1870 a(\)) p 3572 1870 a(at) p 3696 1870 a(most) 0 2009 y(four) p Fn 203 2009 a(p) p Fo('s.) p 388 2009 a(In) p 510 2009 a(the) p 678 2009 a(second) p 993 2009 a(inequalit) m(y) p 1406 2009 a(,) p 1464 2009 a(w) m(e) p 1608 2009 a(used) p 1831 2009 a(\(8.7\)) p 2064 2009 a(and) p Fn 2254 2009 a(\017) p Fo 2321 2009 a(=) p Fn 2424 2009 a(e) p Fg 2469 1972 a(\000) p Fd 2534 1945 a(1) p 2534 1957 31 4 v 2534 1998 a(2) p Fm 2575 1972 a(cR) p Fd 2659 1948 a(2) p Fo 2697 2009 a(.) p 2768 1941 64 4 v 2768 2034 4 93 v 2828 2034 V 2768 2037 64 4 v 146 2222 a(W) p 238 2222 a(e) p 304 2222 a(shall) p 523 2222 a(con) m(tin) m(ue) p 903 2222 a(the) p 1061 2222 a(pro) s(of) p 1306 2222 a(of) p 1407 2222 a(\(i\)) p 1533 2222 a(of) p 1634 2222 a(Theorem) p 2036 2222 a(1.3.) p 2228 2222 a(T) p 2290 2222 a(ak) m(e) p Fn 2453 2222 a(\021) p Fl 2501 2237 a(0) p Fk 2569 2222 a(2) p Fn 2663 2222 a(C) p Fg 2740 2186 a(1) p Fl 2733 2246 a(0) p Fo 2814 2222 a(\() p Fj(R) p Fl 2940 2180 a(2) p Fo 2979 2222 a(\)) p 3040 2222 a(suc) m(h) p 3250 2222 a(that) p 3451 2222 a(0) p Fk 3528 2222 a(\024) p Fn 3633 2222 a(\021) p Fl 3681 2237 a(0) p Fk 3748 2222 a(\024) p Fo 3853 2222 a(1) 0 2342 y(and) p Fn 1029 2586 a(\021) p Fl 1077 2601 a(0) p Fo 1117 2586 a(\() p Fn(z) p Fo 1204 2586 a(\)) p 1270 2586 a(=) p Fi 1374 2440 a(\() p Fo 1457 2526 a(1) p Fn 1604 2526 a(z) p Fk 1681 2526 a(2) p 1775 2526 a(K) p Fl 1851 2541 a(0) p Fo 1890 2526 a(\() p Fm 1947 2487 a(R) p 1938 2503 71 4 v Fl 1938 2561 a(99) p Fo 2019 2526 a(\)) 1457 2647 y(0) p Fn 1604 2647 a(z) p 1693 2647 a(=) p Fk 1681 2647 a(2) p 1775 2647 a(K) p Fl 1851 2662 a(0) p Fo 1890 2647 a(\() p Fm 1947 2607 a(R) p 1938 2623 V Fl 1938 2681 a(98) p Fo 2019 2647 a(\)) p Fn 2116 2586 a(;) p Fo 1029 2854 a(sup) p Fk 1193 2854 a(jr) p Fn(\021) p Fl 1352 2869 a(0) p Fk 1391 2854 a(j) p Fl 1419 2813 a(2) p Fo 1480 2854 a(+) p 1578 2854 a(sup) p Fk 1742 2854 a(jr) p Fn(@) p Fm 1904 2869 a(x) p Fn 1948 2854 a(\021) p Fl 1996 2869 a(0) p Fk 2035 2854 a(j) p Fl 2063 2813 a(2) p Fo 2125 2854 a(+) p 2223 2854 a(sup) p Fk 2386 2854 a(jr) p Fn(@) p Fm 2548 2869 a(y) p Fn 2590 2854 a(\021) p Fl 2638 2869 a(0) p Fk 2677 2854 a(j) p Fl 2705 2813 a(2) p Fk 2772 2854 a(\024) p Fn 2906 2787 a(C) p 2887 2831 115 4 v 2887 2923 a(R) p Fl 2962 2894 a(2) p Fn 3012 2854 a(;) p Fo 3653 2854 a(\(8.15\)) 0 3087 y(where) p Fn 282 3087 a(C) p Fo 391 3087 a(is) p 489 3087 a(a) p 571 3087 a(p) s(ositiv) m(e) p 932 3087 a(constan) m(t) p 1325 3087 a(indep) s(enden) m(t) p 1878 3087 a(of) p Fn 1989 3087 a(R) p Fo 2097 3087 a(and) p Fk 1291 3293 a(K) p Fl 1367 3308 a(0) p Fo 1406 3293 a(\() p Fn(\016) p Fo 1491 3293 a(\)) p 1557 3293 a(=) p Fi 1821 3210 a([) p Fm 1660 3398 a(\014) s(;\016) p Fg 1756 3398 a(2f\000) p Fl(1) p Fm(;) p Fl(0) p Fm(;) p Fl(1) p Fg(g) p Fn 2091 3293 a(K) p Fo 2181 3293 a(\(\() p Fn(\014) p 2318 3293 a(;) p 2362 3293 a(\015) p Fo 2418 3293 a(\);) p Fn 2500 3293 a(\016) p Fo 2547 3293 a(\)) p Fn(:) p Fo 0 3593 a(Put) p Fn 183 3593 a(\021) p Fm 231 3608 a(p) p Fe 267 3618 a(j) p Fo 304 3593 a(\() p Fn(z) p Fo 391 3593 a(\)) p 457 3593 a(=) p Fn 560 3593 a(\021) p Fl 608 3608 a(0) p Fo 648 3593 a(\() p Fn(z) p Fk 742 3593 a(\000) p Fn 825 3593 a(p) p Fm 874 3608 a(j) p Fo 911 3593 a(\)) p 974 3593 a(for) p Fn 1115 3593 a(j) p Fo 1189 3593 a(=) p 1292 3593 a(1) p Fn(;) p 1385 3593 a(:) p 1429 3593 a(:) p 1473 3593 a(:) p 1516 3593 a(;) p 1560 3593 a(N) p Fo 1648 3593 a(.) p 1716 3593 a(By) p 1861 3593 a(Lemma) p 2202 3593 a(8.2,) p 2380 3593 a(w) m(e) p 2516 3593 a(ha) m(v) m(e) p Fn 2733 3593 a(\021) p Fm 2781 3608 a(p) p Fe 2817 3618 a(j) p Fn 2853 3593 a( ) p Fm 2916 3608 a(\017) p Fk 2977 3593 a(2) p Fn 3071 3593 a(D) p Fo 3155 3593 a(\() p Fn(H) p Fm 3274 3608 a(N) p Fo 3341 3593 a(\),) p Fn 3432 3593 a(t) p Fg 3467 3608 a(\000) p Fm(z) p Fe 3555 3618 a(j) p Fn 3591 3593 a(\021) p Fm 3639 3608 a(p) p Fe 3675 3618 a(j) p Fn 3712 3593 a( ) p Fm 3775 3608 a(\017) p Fk 3836 3593 a(2) p Fn 0 3713 a(D) p Fo 84 3713 a(\() p Fn(H) p Fl 203 3728 a(1) p Fo 242 3713 a(\)) p 312 3713 a(and) 924 3920 y(\() p Fn(H) p Fl 1043 3935 a(1) p Fk 1104 3920 a(\000) p Fn 1204 3920 a(E) p Fo 1282 3920 a(\)) p Fn(t) p Fg 1355 3935 a(\000) p Fm(z) p Fe 1443 3945 a(j) p Fn 1479 3920 a(\021) p Fm 1527 3935 a(p) p Fe 1563 3945 a(j) p Fn 1600 3920 a( ) p Fm 1663 3935 a(\017) p Fo 1779 3920 a(=) p Fn 1938 3920 a(t) p Fg 1973 3935 a(\000) p Fm(z) p Fe 2061 3945 a(j) p Fo 2097 3920 a(\() p Fn(H) p Fm 2216 3935 a(N) p Fk 2305 3920 a(\000) p Fn 2405 3920 a(E) p Fo 2483 3920 a(\)\() p Fn(\021) p Fm 2607 3935 a(p) p Fe 2643 3945 a(j) p Fn 2679 3920 a( ) p Fm 2742 3935 a(\017) p Fo 2775 3920 a(\)) 1779 4065 y(=) p Fn 1938 4065 a(t) p Fg 1973 4080 a(\000) p Fm(z) p Fe 2061 4090 a(j) p Fo 2097 4065 a(\([) p Fn(H) p Fm 2243 4080 a(N) p Fn 2310 4065 a(;) p 2354 4065 a(\021) p Fm 2402 4080 a(p) p Fe 2438 4090 a(j) p Fo 2474 4065 a(]) p Fn( ) p Fm 2564 4080 a(\017) p Fo 2620 4065 a(+) p Fn 2718 4065 a(\021) p Fm 2766 4080 a(p) p Fe 2802 4090 a(j) p Fn 2838 4065 a(\030) p Fm 2881 4080 a(\017) p Fo 2913 4065 a(\)) p Fn(:) p Fo 3653 4065 a(\(8.16\)) 0 4287 y(Let) p Fn 175 4287 a(c) p Fo 249 4287 a(b) s(e) p 382 4287 a(a) p 463 4287 a(constan) m(t) p 857 4287 a(giv) m(en) p 1111 4287 a(in) p 1225 4287 a(Lemma) p 1573 4287 a(8.5) p 1731 4287 a(and) p 1920 4287 a(put) p Fn 2099 4287 a(\017) p Fo 2166 4287 a(=) p Fn 2270 4287 a(e) p Fg 2315 4250 a(\000) p Fe 2381 4223 a(c) p 2380 4235 31 4 v Fd 2380 4276 a(2) p Fm 2420 4250 a(R) p Fd 2473 4226 a(2) p Fo 2513 4287 a(.) p 2583 4287 a(Th) m(us) p 2830 4287 a(w) m(e) p 2974 4287 a(ha) m(v) m(e) p Fm 821 4454 a(N) p Fi 792 4479 a(X) p Fm 791 4661 a(j) p Fl 824 4661 a(=1) p Fk 930 4562 a(jj) p Fo(\() p Fn(H) p Fl 1105 4577 a(1) p Fk 1166 4562 a(\000) p Fn 1265 4562 a(E) p Fo 1343 4562 a(\)) p Fn(t) p Fg 1416 4577 a(\000) p Fm(z) p Fe 1504 4587 a(j) p Fn 1541 4562 a(\021) p Fm 1589 4577 a(p) p Fe 1625 4587 a(j) p Fn 1661 4562 a( ) p Fm 1724 4577 a(\017) p Fk 1757 4562 a(jj) p Fl 1813 4521 a(2) p Fk 630 4877 a(\024) p Fo 791 4877 a(2) p Fm 886 4770 a(N) p Fi 858 4794 a(X) p Fm 856 4977 a(j) p Fl 889 4977 a(=1) p Fi 995 4781 a(\020) p Fk 1045 4877 a(j) p Fo([) p Fn(H) p Fm 1181 4892 a(N) p Fn 1248 4877 a(;) p 1292 4877 a(\021) p Fm 1340 4892 a(p) p Fe 1376 4902 a(j) p Fo 1412 4877 a(]) p Fn( ) p Fm 1502 4892 a(\017) p Fk 1535 4877 a(jj) p Fl 1591 4836 a(2) p Fo 1652 4877 a(+) p Fk 1750 4877 a(jj) p Fn(\021) p Fm 1854 4892 a(p) p Fe 1890 4902 a(j) p Fn 1926 4877 a(\030) p Fm 1969 4892 a(\017) p Fk 2001 4877 a(jj) p Fl 2057 4836 a(2) p Fi 2096 4781 a(\021) p Fo 1902 5214 a(50) p eop %%Page: 51 51 51 50 bop Fk 630 693 a(\024) p Fo 791 693 a(2\() p Fn(C) p Fl 948 708 a(5) p Fo 1009 693 a(+) p 1107 693 a(1\)) p Fm 1240 585 a(N) p Fi 1212 610 a(X) p Fm 1210 792 a(j) p Fl 1243 792 a(=1) p Fi 1349 547 a( ) 1415 576 y(Z) p 1461 693 484 4 v Fg 1461 764 a(K) p Fe 1515 774 a(j) p Fl 1547 764 a(\() p Fe 1592 737 a(R) p 1585 749 61 4 v Fd 1585 790 a(98) p Fl 1656 764 a(\)) p Fg(nK) p Fe 1772 774 a(j) p Fl 1805 764 a(\() p Fe 1849 737 a(R) p 1842 749 V Fd 1842 790 a(99) p Fl 1913 764 a(\)) p Fn 1961 693 a(dxdy) p Fk 2170 693 a(j) p Fn( ) p Fm 2261 708 a(\017) p Fo 2293 693 a(\() p Fn(z) p Fo 2380 693 a(\)) p Fk(j) p Fl 2446 652 a(2) p Fo 2508 693 a(+) p Fi 2606 576 a(Z) p Fg 2652 764 a(K) p Fe 2706 774 a(j) p Fl 2738 764 a(\() p Fe 2783 737 a(R) p 2775 749 V Fd 2775 790 a(98) p Fl 2846 764 a(\)) p Fn 2894 693 a(dxdy) p Fk 3103 693 a(j) p Fn(\030) p Fm 3174 708 a(\017) p Fk 3205 693 a(j) p Fl 3233 652 a(2) p Fi 3272 547 a(!) p Fk 630 996 a(\024) p Fn 791 996 a(C) p Fi 884 850 a( ) 950 879 y(Z) p Fg 996 1068 a(F) p Fe 1053 1045 a(c) 1046 1092 y(N) p Fn 1125 996 a(dxdy) p Fk 1334 996 a(j) p Fn( ) p Fm 1425 1011 a(\017) p Fo 1457 996 a(\() p Fn(z) p Fo 1544 996 a(\)) p Fk(j) p Fl 1610 955 a(2) p Fo 1672 996 a(+) p Fn 1770 996 a(\017) p Fl 1809 955 a(2) p Fi 1849 850 a(!) p Fk 1942 996 a(\024) p Fn 2047 996 a(C) p 2124 996 a(e) p Fg 2169 955 a(\000) p Fm(cR) p Fd 2308 932 a(2) p Fn 2347 996 a(;) p Fo 2423 996 a(if) p Fn 2520 996 a(R) p Fk 2622 996 a(\025) p Fo 2727 996 a(1) p Fn(;) p Fo 3653 996 a(\(8.17\)) 0 1278 y(where) p Fn 282 1278 a(C) p Fo 391 1278 a(is) p 490 1278 a(a) p 571 1278 a(constan) m(t) p 965 1278 a(dep) s(enden) m(t) p 1436 1278 a(only) p 1650 1278 a(on) p Fn 1786 1278 a(B) p Fo 1865 1278 a(,) p Fn 1925 1278 a(E) p Fo 2003 1278 a(.) p 2074 1278 a(W) p 2166 1278 a(e) p 2242 1278 a(used) p 2465 1278 a(\(8.16\)) p 2747 1278 a(in) p 2861 1278 a(the) p 3029 1278 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a(2) p Fm 1938 1901 a(N) p Fi 1910 1925 a(X) p Fm 1909 2108 a(j) p Fl 1942 2108 a(=1) p Fk 2048 2008 a(jj) p Fn(\021) p Fm 2152 2023 a(p) p Fe 2188 2033 a(j) p Fn 2224 2008 a( ) p Fm 2287 2023 a(\017) p Fk 2320 2008 a(jj) p Fl 2376 1967 a(2) p Fk 1058 2279 a(\025) p Fo 1218 2279 a(dist) p Fk(f) p Fn(E) p 1504 2279 a(;) p 1548 2279 a(\033) p Fo 1607 2279 a(\() p Fn(H) p Fl 1726 2294 a(1) p Fo 1765 2279 a(\)) p Fk(g) p Fl 1853 2238 a(2) p Fi 1909 2162 a(Z) p Fg 1955 2351 a(F) p Fe 2005 2362 a(N) p Fn 2084 2279 a(dxdy) p Fk 2293 2279 a(j) p Fn( ) p Fm 2384 2294 a(\017) p Fk 2415 2279 a(j) p Fl 2443 2238 a(2) p Fk 1058 2492 a(\025) p Fo 1218 2492 a(dist) p Fk(f) p Fn(E) p 1504 2492 a(;) p 1548 2492 a(\033) p Fo 1607 2492 a(\() p Fn(H) p Fl 1726 2507 a(1) p Fo 1765 2492 a(\)) p Fk(g) p Fl 1853 2451 a(2) p Fo 1892 2492 a(\(1) p Fk 2001 2492 a(\000) p Fn 2101 2492 a(C) p Fl 2171 2507 a(7) p Fn 2210 2492 a(e) p Fg 2255 2451 a(\000) p Fm(cR) p Fd 2394 2427 a(2) p Fo 2433 2492 a(\)) p Fn(;) p Fo 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Fl 1605 3086 a(1) p Fo 1644 3071 a(\)) p Fk(g) p 1759 3071 a(\024) 1864 2983 y(p) p 1947 2983 126 4 v Fo 1947 3071 a(2) p Fn(C) p 2073 3071 a(e) p Fg 2118 3030 a(\000) p Fe 2184 3003 a(c) p 2183 3015 31 4 v Fd 2183 3056 a(2) p Fm 2224 3030 a(R) p Fd 2277 3007 a(2) p Fn 2316 3071 a(;) p Fo 2360 3071 a(if) p Fn 2456 3071 a(R) p Fk 2559 3071 a(\025) p Fn 2664 3071 a(R) p Fl 2738 3086 a(0) p Fn 2778 3071 a(:) p Fo 0 3291 a(Th) m(us) p 247 3291 a(\(i\)) p 383 3291 a(of) p 494 3291 a(Theorem) p 906 3291 a(1.3) p 1063 3291 a(holds.) p 1356 3224 64 4 v 1356 3316 4 93 v 1416 3316 V 1356 3319 64 4 v Ff 146 3520 a(Pr) p 248 3520 a(o) p 293 3520 a(of) p 406 3520 a(of) p 518 3520 a(\(ii\)) p 690 3520 a(of) p 802 3520 a(The) p 962 3520 a(or) p 1048 3520 a(em) p 1205 3520 a(1.3.) p Fo 1437 3520 a(Let) p Fn 1609 3520 a(I) p Fo 1690 3520 a(b) s(e) p 1820 3520 a(an) p 1953 3520 a(in) m(terv) p 2197 3520 a(al) p 2303 3520 a(satisfying) p 2735 3520 a(the) p 2901 3520 a(assumption) p 3414 3520 a(of) p 3523 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1674 4438 a(exists) p Fn 1943 4438 a(R) p Fl 2017 4453 a(1) p Fn 2085 4438 a(>) p 2188 4438 a(R) p Fl 2262 4453 a(0) p Fo 2335 4438 a(suc) m(h) p 2555 4438 a(that) 1009 4658 y(dim) p 1188 4658 a(Ran) p Fn 1380 4658 a(P) p Fl 1443 4674 a(\() p Fm(E) p Fg 1526 4674 a(\000) p Fm(\016) p Fd 1612 4683 a(0) p Fm 1646 4674 a(;E) p Fl 1722 4674 a(+) p Fm(\016) p Fd 1808 4683 a(0) p Fl 1842 4674 a(\)) p Fo 1874 4658 a(\() p Fn(H) p Fm 1993 4673 a(N) p Fo 2060 4658 a(\)) p 2125 4658 a(=) p Fn 2229 4658 a(N) p 2317 4658 a(k) s(;) p Fo 2448 4658 a(if) p Fn 2544 4658 a(R) p Fk 2647 4658 a(\025) p Fn 2752 4658 a(R) p Fl 2826 4673 a(1) p Fn 2866 4658 a(;) p Fo 0 4879 a(where) p Fn 282 4879 a(k) p Fo 364 4879 a(=) p 467 4879 a(m) m(ult) o(\() p Fn(E) p Fo 781 4879 a(;) p Fn 825 4879 a(H) p Fl 906 4894 a(1) p Fo 945 4879 a(\).) 1902 5214 y(51) p eop %%Page: 52 52 52 51 bop Fo 146 631 a(First) p 384 631 a(w) m(e) p 528 631 a(sho) m(w) p 769 631 a(there) p 1019 631 a(exists) p Fn 1288 631 a(R) p Fl 1362 646 a(1) p Fn 1429 631 a(>) p 1533 631 a(R) p Fl 1607 646 a(0) p Fo 1679 631 a(suc) m(h) p 1899 631 a(that) 1269 851 y(dim) p 1448 851 a(Ran) p Fn 1640 851 a(P) p Fl 1703 866 a(\() p Fm(E) p Fg 1786 866 a(\000) p Fm(\016) p Fd 1872 875 a(0) p Fm 1906 866 a(;E) p Fl 1982 866 a(+) p Fm(\016) p Fd 2068 875 a(0) p Fl 2102 866 a(\)) p Fo 2133 851 a(\() p Fn(H) p Fm 2252 866 a(N) p Fo 2319 851 a(\)) p Fk 2385 851 a(\025) p Fn 2490 851 a(N) p 2578 851 a(k) p Fo 3653 851 a(\(8.19\)) 0 1071 y(for) p 152 1071 a(eac) m(h) p 374 1071 a(eigen) m(v) p 634 1071 a(alue) p Fn 843 1071 a(E) p Fo 956 1071 a(of) p Fn 1070 1071 a(H) p Fl 1151 1086 a(1) p Fo 1225 1071 a(con) m(tained) p 1669 1071 a(in) p Fn 1786 1071 a(I) p Fo 1837 1071 a(,) p 1900 1071 a(if) p Fn 1992 1071 a(R) p 2099 1071 a(>) p 2207 1071 a(R) p Fl 2281 1086 a(1) p Fo 2321 1071 a(.) p 2399 1071 a(Let) p Fk 2577 1071 a(f) p Fn(e) p Fl 2672 1086 a(1) p Fn 2711 1071 a(;) p 2755 1071 a(:) p 2799 1071 a(:) p 2843 1071 a(:) p 2886 1071 a(;) p 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2856 a(ha) m(v) m(e) p Fn 1175 2856 a(V) p Fk 1281 2856 a(\032) p Fn 1386 2856 a(D) p Fo 1470 2856 a(\() p Fn(H) p Fm 1589 2871 a(N) p Fo 1656 2856 a(\).) p 1764 2856 a(Applying) p 2187 2856 a(Lemma) p 2535 2856 a(8.6) p 2692 2856 a(to) p 2811 2856 a(the) p 2979 2856 a(case) p Fn 3186 2856 a(N) p Fo 3302 2856 a(=) p 3406 2856 a(1,) p 3514 2856 a(w) m(e) p 3658 2856 a(ha) m(v) m(e) p Fi 1111 2989 a(Z) p Fm 1157 3178 a(K) p Fl 1222 3178 a(\(0) p Fm(;) p Fe 1336 3151 a(R) p 1314 3163 92 4 v Fd 1314 3204 a(100) p Fl 1415 3178 a(\)) p Fe 1442 3159 a(c) p Fn 1495 3106 a(dxdy) p Fk 1704 3106 a(j) p Fn(e) p Fm 1777 3121 a(l) p Fk 1802 3106 a(j) p Fl 1830 3065 a(2) p Fk 1897 3106 a(\024) p Fn 2002 3106 a(C) p Fl 2072 3121 a(7) p Fn 2111 3106 a(e) p Fg 2156 3065 a(\000) p Fm(cR) p Fd 2295 3042 a(2) p Fn 2334 3106 a(;) p Fo 2411 3106 a(if) p Fn 2507 3106 a(R) p Fk 2610 3106 a(\025) p Fo 2715 3106 a(1) p Fn(;) p Fo 3653 3106 a(\(8.20\)) 0 3381 y(where) p Fn 282 3381 a(c) p Fo 356 3381 a(and) p Fn 546 3381 a(C) p Fl 616 3396 a(7) p Fo 688 3381 a(is) p 786 3381 a(a) p 867 3381 a(constan) m(t) p 1261 3381 a(dep) s(enden) m(t) p 1732 3381 a(only) p 1946 3381 a(on) p Fn 2082 3381 a(B) p Fo 2161 3381 a(,) p Fn 2220 3381 a(E) p Fo 2298 3381 a(.) p 2369 3381 a(By) p 2522 3381 a(the) p 2690 3381 a(equalit) m(y) 418 3601 y(\() t(~) p Fn 456 3601 a(e) p Fm 501 3616 a(j;l) p Fn 576 3601 a(;) p Fo 623 3601 a(~) p Fn 620 3601 a(e) p Fm 665 3616 a(j;l) p Fb 736 3597 a(0) p Fo 761 3601 a(\)) p 827 3601 a(=) p 930 3601 a(\() p Fn(\021) p Fl 1016 3616 a(0) p Fn 1056 3601 a(e) p Fm 1101 3616 a(l) p Fn 1127 3601 a(;) p 1171 3601 a(\021) p Fl 1219 3616 a(0) p Fn 1258 3601 a(e) p Fm 1303 3616 a(l) p Fb 1325 3597 a(0) p Fo 1352 3601 a(\)) p 1473 3601 a(=) p 1631 3601 a(\() p Fn(e) p Fm 1714 3616 a(l) p Fn 1741 3601 a(;) p 1785 3601 a(e) p Fm 1830 3616 a(l) p Fb 1852 3597 a(0) p Fo 1878 3601 a(\)) p Fk 1938 3601 a(\000) p Fo 2038 3601 a(\(\(1) p Fk 2184 3601 a(\000) p Fn 2284 3601 a(\021) p Fl 2332 3616 a(0) p Fo 2372 3601 a(\)) p Fn(e) p Fm 2455 3616 a(l) p Fn 2481 3601 a(;) p 2525 3601 a(e) p Fm 2570 3616 a(l) p Fb 2592 3597 a(0) p Fo 2618 3601 a(\)) p Fk 2678 3601 a(\000) p Fo 2778 3601 a(\() p Fn(\021) p Fl 2864 3616 a(0) p Fn 2903 3601 a(e) p Fm 2948 3616 a(l) p Fn 2974 3601 a(;) p Fo 3018 3601 a(\(1) p Fk 3127 3601 a(\000) p Fn 3227 3601 a(\021) p Fl 3275 3616 a(0) p Fo 3314 3601 a(\)) p Fn(e) p Fm 3397 3616 a(l) p Fb 3419 3597 a(0) p Fo 3446 3601 a(\)) 1473 3746 y(=) p Fn 1631 3746 a(\016) p Fm 1674 3761 a(l) q(;l) p Fb 1738 3742 a(0) p Fk 1787 3746 a(\000) p Fo 1886 3746 a(\(\(1) p Fk 2033 3746 a(\000) p Fn 2133 3746 a(\021) p Fl 2181 3761 a(0) p Fo 2220 3746 a(\)) p Fn(e) p Fm 2303 3761 a(l) p Fn 2329 3746 a(;) p 2373 3746 a(e) p Fm 2418 3761 a(l) p Fb 2440 3742 a(0) p Fo 2467 3746 a(\)) p Fk 2527 3746 a(\000) p Fo 2626 3746 a(\() p Fn(\021) p Fl 2712 3761 a(0) p Fn 2752 3746 a(e) p Fm 2797 3761 a(l) p Fn 2823 3746 a(;) p Fo 2867 3746 a(\(1) p Fk 2976 3746 a(\000) p Fn 3075 3746 a(\021) p Fl 3123 3761 a(0) p Fo 3163 3746 a(\)) p Fn(e) p Fm 3246 3761 a(l) p Fb 3268 3742 a(0) p Fo 3294 3746 a(\)) p 3653 3746 a(\(8.21\)) 0 3966 y(and) p 190 3966 a(\(8.20\),) p 499 3966 a(w) m(e) p 642 3966 a(ha) m(v) m(e) p Fk 1145 4096 a(j) p Fo(\() t(~) p Fn 1211 4096 a(e) p Fm 1256 4111 a(j;l) p Fn 1330 4096 a(;) p Fo 1377 4096 a(~) p Fn 1374 4096 a(e) p Fm 1419 4111 a(j;l) p Fb 1490 4092 a(0) p Fo 1515 4096 a(\)) p Fk 1575 4096 a(\000) p Fn 1675 4096 a(\016) p Fm 1718 4111 a(l) q(;l) p Fb 1782 4092 a(0) p Fk 1808 4096 a(j) p 1863 4096 a(\024) p Fn 1968 4096 a(C) p Fl 2038 4111 a(8) p Fn 2078 4096 a(e) p Fg 2123 4055 a(\000) p Fm(cR) p Fd 2262 4032 a(2) p Fn 2301 4096 a(;) p Fo 2377 4096 a(if) p Fn 2473 4096 a(R) p Fk 2576 4096 a(\025) p Fo 2681 4096 a(1) p Fn(;) p Fo 3653 4096 a(\(8.22\)) 0 4271 y(where) p Fn 283 4271 a(C) p Fl 353 4286 a(8) p Fo 426 4271 a(is) p 525 4271 a(a) p 608 4271 a(constan) m(t) p 1002 4271 a(dep) s(enden) m(t) p 1475 4271 a(only) p 1690 4271 a(on) p Fn 1827 4271 a(B) p Fo 1906 4271 a(,) p Fn 1967 4271 a(E) p Fo 2045 4271 a(.) p 2120 4271 a(Let) p Fn 2295 4271 a(M) p Fo 2430 4271 a(=) p 2536 4271 a(\() p Fn(m) p Fm 2659 4286 a(l) q(;l) p Fb 2723 4267 a(0) p Fo 2749 4271 a(\)) p Fm 2787 4286 a(l) q(;l) p Fb 2851 4267 a(0) p Fl 2872 4286 a(=1) p Fm(;:::) o(;k) p Fo 3137 4271 a(b) s(e) p 3271 4271 a(a) p Fn 3353 4271 a(k) p Fk 3430 4271 a(\002) p Fn 3531 4271 a(k) p Fo 3585 4271 a(-matrix) 0 4391 y(giv) m(en) p 250 4391 a(b) m(y) p Fn 381 4391 a(m) p Fm 466 4406 a(l) q(;l) p Fb 530 4387 a(0) p Fo 584 4391 a(=) p 688 4391 a(\() t(~) p Fn 726 4391 a(e) p Fm 771 4406 a(j;l) p Fn 845 4391 a(;) p Fo 892 4391 a(~) p Fn 889 4391 a(e) p Fm 934 4406 a(j;l) p Fb 1005 4387 a(0) p Fo 1030 4391 a(\).) p 1137 4391 a(By) p 1286 4391 a(\(8.21\),) p 1591 4391 a(w) m(e) p 1731 4391 a(see) p 1884 4391 a(that) p Fn 2091 4391 a(M) p Fo 2224 4391 a(is) p 2317 4391 a(a) p 2394 4391 a(matrix) p 2707 4391 a(indep) s(enden) m(t) p 3255 4391 a(of) p Fn 3362 4391 a(j) p Fo 3408 4391 a(.) p 3477 4391 a(By) p 3626 4391 a(\(8.22\),) 0 4511 y(w) m(e) p 144 4511 a(ha) m(v) m(e) p 369 4511 a(that) p 580 4511 a(there) p 829 4511 a(exists) p 1099 4511 a(a) p 1180 4511 a(constan) m(t) p Fn 1573 4511 a(R) p Fl 1647 4526 a(1) p Fn 1715 4511 a(>) p 1818 4511 a(R) p Fl 1892 4526 a(0) p Fo 1964 4511 a(dep) s(enden) m(t) p 2436 4511 a(only) p 2650 4511 a(on) p Fn 2785 4511 a(B) p Fo 2864 4511 a(,) p Fn 2924 4511 a(E) p Fo 3035 4511 a(suc) m(h) p 3255 4511 a(that) p Fk 926 4770 a(j) p Fn(m) p Fm 1039 4785 a(l) q(;l) p Fb 1103 4766 a(0) p Fk 1129 4770 a(j) p Fo 1184 4770 a(=) p Fk 1288 4770 a(j) p Fo(\() t(~) p Fn 1354 4770 a(e) p Fm 1399 4785 a(j;l) p Fn 1473 4770 a(;) p Fo 1520 4770 a(~) p Fn 1517 4770 a(e) p Fm 1562 4785 a(j;l) p Fb 1633 4766 a(0) p Fo 1658 4770 a(\)) p Fk(j) p 1751 4770 a(\024) p Fo 1856 4770 a(2) p Fn(;) p Fo 1998 4770 a(det) p Fn 2150 4770 a(M) p 2282 4770 a(>) p Fo 2396 4703 a(1) p 2396 4747 49 4 v 2396 4839 a(2) p Fn 2455 4770 a(;) p Fo 2531 4770 a(if) p Fn 2627 4770 a(R) p Fk 2730 4770 a(\025) p Fn 2835 4770 a(R) p Fl 2909 4785 a(1) p Fn 2949 4770 a(:) p Fo 3653 4770 a(\(8.23\)) 1902 5214 y(52) p eop %%Page: 53 53 53 52 bop Fo 0 631 a(This) p 219 631 a(inequalit) m(y) p 667 631 a(implies) p 994 631 a(that) p Fn 1201 631 a(M) p Fo 1335 631 a(is) p 1429 631 a(in) m(v) m(ertible) p 1856 631 a(and) p 2041 631 a(the) p 2205 631 a(v) m(ectors) p Fk 2532 631 a(f) p Fo 2586 631 a(~) p Fn 2582 631 a(e) p Fm 2627 646 a(j;l) p Fk 2701 631 a(g) p Fm 2751 646 a(l) p Fl 2773 646 a(=1) p Fm(;:::) n(;k) p Fo 3032 631 a(are) p 3191 631 a(linearly) p 3536 631 a(indep) s(en-) 0 751 y(den) m(t,) p 247 751 a(if) p Fn 336 751 a(R) p Fk 439 751 a(\025) p Fn 544 751 a(R) p Fl 618 766 a(1) p Fo 658 751 a(.) p 728 751 a(Th) m(us) 1400 941 y(dim) p Fn 1579 941 a(V) p Fo 1686 941 a(=) p Fm 1819 833 a(N) p Fi 1790 858 a(X) p Fm 1789 1040 a(j) p Fl 1822 1040 a(=1) p Fo 1928 941 a(dim) p Fn 2107 941 a(V) p Fm 2164 956 a(j) p Fo 2229 941 a(=) p Fn 2332 941 a(N) p 2420 941 a(k) s(:) p Fo 3653 941 a(\(8.24\)) 0 1193 y(Let) p Fn 175 1193 a(P) p Fm 238 1208 a(j) p Fo 307 1193 a(b) s(e) p 439 1193 a(the) p 607 1193 a(orthogonal) p 1095 1193 a(pro) p 1241 1193 a(jection) p 1559 1193 a(on) m(to) p Fn 1778 1193 a(V) p Fm 1835 1208 a(j) p Fo 1904 1193 a(and) p 2094 1193 a(put) p Fn 1622 1491 a(P) p Fm 1685 1506 a(j) p Fn 1721 1491 a(u) p Fo 1805 1491 a(=) p Fm 1949 1383 a(k) p Fi 1908 1408 a(X) p Fm 1912 1592 a(l) p Fl 1934 1592 a(=1) p Fn 2045 1491 a(c) p Fm 2087 1506 a(j;l) p Fo 2164 1491 a(~) p Fn 2161 1491 a(e) p Fm 2206 1506 a(j;l) p Fo 0 1793 a(for) p Fn 149 1793 a(u) p Fk 232 1793 a(2) p Fn 326 1793 a(L) p Fl 392 1757 a(2) p Fo 432 1793 a(\() p Fj(R) p Fl 557 1751 a(2) p Fo 597 1793 a(\).) p 705 1793 a(The) p 906 1793 a(co) s(e\016cien) m(t) p Fn 1361 1793 a(c) p Fm 1403 1808 a(j;l) p Fo 1510 1793 a(is) p 1608 1793 a(written) p 1949 1793 a(as) 897 2013 y(\() p Fn(c) p Fm 977 2028 a(j;) p Fl(1) p Fn 1065 2013 a(;) p 1109 2013 a(:) p 1153 2013 a(:) p 1197 2013 a(:) p 1239 2013 a(;) p 1283 2013 a(c) p Fm 1325 2028 a(j;k) p Fo 1416 2013 a(\)) p 1481 2013 a(=) p 1585 2013 a(\() p 1655 2013 a(\() p Fn(P) p Fm 1756 2028 a(j) p Fn 1793 2013 a(u;) p 1893 2013 a(e) p Fm 1938 2028 a(j;) p Fl(1) p Fo 2025 2013 a(\)) p Fn(;) p 2107 2013 a(:) p 2151 2013 a(:) p 2195 2013 a(:) p 2237 2013 a(;) p Fo 2281 2013 a(\() p Fn(P) p Fm 2382 2028 a(j) p Fn 2418 2013 a(u;) p 2518 2013 a(e) p Fm 2563 2028 a(j;k) p Fo 2654 2013 a(\)) p 2724 2013 a(\)) p Fn 2779 2013 a(M) p Fg 2883 1972 a(\000) p Fl(1) p Fn 2978 2013 a(;) p Fo 0 2233 a(where) p 285 2233 a(the) p 456 2233 a(righ) m(t) p 695 2233 a(hand) p 942 2233 a(side) p 1140 2233 a(is) p 1241 2233 a(a) p 1326 2233 a(pro) s(duct) p 1695 2233 a(of) p 1809 2233 a(a) p 1893 2233 a(column) p 2237 2233 a(v) m(ector) p 2533 2233 a(and) p 2726 2233 a(a) p 2810 2233 a(matrix.) p 3174 2233 a(Th) m(us) p 3424 2233 a(w) m(e) p 3571 2233 a(ha) m(v) m(e) p 3799 2233 a(b) m(y) 0 2354 y(\(8.23\)) p Fm 1282 2441 a(k) p Fi 1242 2466 a(X) p Fm 1245 2651 a(l) p Fl 1267 2651 a(=1) p Fk 1378 2549 a(j) p Fn(c) p Fm 1448 2564 a(j;l) p Fk 1522 2549 a(j) p Fl 1550 2508 a(2) p Fk 1617 2549 a(\024) p Fn 1722 2549 a(C) p Fm 1792 2564 a(k) p Fk 1834 2549 a(jj) p Fn(P) p Fm 1953 2564 a(j) p Fn 1989 2549 a(u) p Fk(jj) p Fl 2101 2508 a(2) p Fn 2139 2549 a(;) p Fo 2215 2549 a(if) p Fn 2312 2549 a(R) p Fk 2415 2549 a(\025) p Fn 2520 2549 a(R) p Fl 2594 2564 a(1) p Fn 2633 2549 a(;) p Fo 0 2790 a(where) p Fn 274 2790 a(C) p Fm 344 2805 a(k) p Fo 412 2790 a(is) p 503 2790 a(a) p 577 2790 a(constan) m(t) p 963 2790 a(dep) s(enden) m (t) p 1427 2790 a(only) p 1634 2790 a(on) p Fn 1762 2790 a(k) p Fo 1816 2790 a(.) p 1884 2790 a(Summing) p 2310 2790 a(up) p 2444 2790 a(this) p 2627 2790 a(inequalit) m(y) p 3072 2790 a(for) p 3214 2790 a(all) p Fn 3342 2790 a(j) p Fo 3416 2790 a(=) p 3519 2790 a(1) p Fn(;) p 3612 2790 a(:) p 3656 2790 a(:) p 3700 2790 a(:) p 3743 2790 a(;) p 3787 2790 a(N) p Fo 3875 2790 a(,) 0 2911 y(w) m(e) p 144 2911 a(ha) m(v) m(e) p Fm 1498 2974 a(N) p Fi 1470 2998 a(X) p Fm 1469 3181 a(j) p Fl 1502 3181 a(=1) p Fm 1649 2974 a(k) p Fi 1608 2998 a(X) p Fm 1612 3183 a(l) p Fl 1634 3183 a(=1) p Fk 1744 3082 a(j) p Fn(c) p Fm 1814 3097 a(j;l) p Fk 1888 3082 a(j) p Fl 1916 3040 a(2) p Fk 1983 3082 a(\024) p Fn 2088 3082 a(C) p Fm 2158 3097 a(k) p Fk 2201 3082 a(jj) p Fn(u) p Fk(jj) p Fl 2369 3040 a(2) p Fn 2406 3082 a(;) p Fo 3653 3082 a(\(8.25\)) 0 3349 y(for) p 149 3349 a(ev) m(ery) p Fn 406 3349 a(u) p Fk 490 3349 a(2) p Fn 584 3349 a(L) p Fl 650 3313 a(2) p Fo 690 3349 a(\() p Fj(R) p Fl 815 3307 a(2) p Fo 854 3349 a(\),) p 952 3349 a(if) p Fn 1041 3349 a(R) p Fk 1144 3349 a(\025) p Fn 1249 3349 a(R) p Fl 1323 3364 a(1) p Fo 1363 3349 a(.) 146 3469 y(Supp) s(ose) p 541 3469 a(that,) p 795 3469 a(for) p 957 3469 a(an) m(y) p 1154 3469 a(p) s(ositiv) m(e) p 1527 3469 a(n) m(um) m(b) s(er) p Fn 1895 3469 a(R) p Fl 1969 3484 a(3) p Fo 2008 3469 a(,) p 2083 3469 a(there) p 2345 3469 a(exists) p 2627 3469 a(a) p 2721 3469 a(sequence) p Fk 3138 3469 a(f) p Fn(z) p Fm 3233 3484 a(j) p Fk 3269 3469 a(g) p Fm 3319 3433 a(N) 3319 3494 y(j) p Fl 3352 3494 a(=1) p Fo 3491 3469 a(suc) m(h) p 3723 3469 a(that) p Fn 0 3590 a(R) p Fo 116 3590 a(=) p 232 3590 a(inf) p Fm 350 3605 a(j) p Fg 383 3605 a(6) p Fl(=) p Fm(j) p Fb 471 3586 a(0) p Fk 513 3590 a(j) p Fn(z) p Fm 586 3605 a(j) p Fk 649 3590 a(\000) p Fn 754 3590 a(z) p Fm 799 3605 a(j) p Fb 832 3586 a(0) p Fk 858 3590 a(j) p 927 3590 a(\025) p Fn 1044 3590 a(R) p Fl 1118 3605 a(3) p Fo 1198 3590 a(and) p 1395 3590 a(dim) p 1575 3590 a(Ran) p Fn 1766 3590 a(P) p Fl 1829 3605 a(\() p Fm(E) p Fg 1912 3605 a(\000) p Fm(\016) p Fd 1998 3614 a(0) p Fm 2032 3605 a(;E) p Fl 2108 3605 a(+) p Fm(\016) p Fd 2194 3614 a(0) p Fl 2228 3605 a(\)) p Fo 2260 3590 a(\() p Fn(H) p Fm 2379 3605 a(N) p Fo 2446 3590 a(\)) p Fk 2524 3590 a(\024) p Fn 2642 3590 a(N) p 2730 3590 a(k) p Fk 2812 3590 a(\000) p Fo 2917 3590 a(1) p 3005 3590 a(hold) p 3230 3590 a(\(when) p Fn 3530 3590 a(N) p Fo 3659 3590 a(=) p Fk 3775 3590 a(1) p Fo(,) 0 3710 y(w) m(e) p 148 3710 a(replace) p 483 3710 a(the) p 655 3710 a(latter) p 925 3710 a(assumption) p 1446 3710 a(b) m(y) p 1586 3710 a(`dim) p 1792 3710 a(Ran) p Fn 1983 3710 a(P) p Fl 2046 3726 a(\() p Fm(E) p Fg 2129 3726 a(\000) p Fm(\016) p Fd 2215 3735 a(0) p Fm 2250 3726 a(;E) p Fl 2326 3726 a(+) p Fm(\016) p Fd 2412 3735 a(0) p Fl 2445 3726 a(\)) p Fo 2477 3710 a(\() p Fn(H) p Fg 2596 3725 a(1) p Fo 2671 3710 a(\)) p 2745 3710 a(is) p 2848 3710 a(\014nite'\).) p 3213 3710 a(By) p 3371 3710 a(\(8.24\),) p 3685 3710 a(there) 0 3870 y(exists) p 279 3870 a(~) p Fn 276 3870 a(e) p Fo 359 3870 a(=) p Fi 474 3804 a(P) p Fm 561 3891 a(j;l) p Fn 652 3870 a(c) p Fm 694 3885 a(j;l) p Fo 772 3870 a(~) p Fn 768 3870 a(e) p Fm 813 3885 a(j;l) p Fk 926 3870 a(2) p Fn 1031 3870 a(V) p Fo 1148 3870 a(suc) m(h) p 1374 3870 a(that) p Fk 1592 3870 a(jj) p Fo 1652 3870 a(~) p Fn 1648 3870 a(e) p Fk(jj) p Fo 1786 3870 a(=) p 1901 3870 a(1) p 1988 3870 a(and) p 2188 3870 a(~) p Fn 2184 3870 a(e) p Fk 2268 3870 a(2) p Fi 2373 3774 a(\020) p Fo 2423 3870 a(Ran) p Fn 2614 3870 a(P) p Fl 2677 3886 a(\() p Fm(E) p Fg 2760 3886 a(\000) p Fm(\016) p Fd 2846 3895 a(0) p Fm 2880 3886 a(;E) p Fl 2956 3886 a(+) p Fm(\016) p Fd 3042 3895 a(0) p Fl 3076 3886 a(\)) p Fo 3108 3870 a(\() p Fn(H) p Fm 3227 3885 a(N) p Fo 3294 3870 a(\)) p Fi 3332 3774 a(\021) p Fg 3381 3797 a(?) p Fo 3440 3870 a(.) p 3530 3870 a(Then) p 3791 3870 a(w) m(e) 0 3990 y(ha) m(v) m(e) p Fk 1540 4111 a(jj) p Fo(\() p Fn(H) p Fm 1715 4126 a(N) p Fk 1803 4111 a(\000) p Fn 1903 4111 a(E) p Fo 1981 4111 a(\)) t(~) p Fn 2019 4111 a(e) p Fk(jj) p 2147 4111 a(\025) p Fn 2252 4111 a(\016) p Fl 2295 4126 a(0) p Fn 2335 4111 a(:) p Fo 3653 4111 a(\(8.26\)) 0 4285 y(But,) p 221 4285 a(w) m(e) p 364 4285 a(ha) m(v) m(e) p 589 4285 a(b) m(y) p 725 4285 a(\(ii\)) p 887 4285 a(of) p 999 4285 a(Lemma) p 1347 4285 a(8.2) 1170 4505 y(\() p Fn(H) p Fm 1289 4520 a(N) p Fk 1378 4505 a(\000) p Fn 1477 4505 a(E) p Fo 1555 4505 a(\)) t(~) p Fn 1593 4505 a(e) p Fo 1666 4505 a(=) p Fi 1770 4422 a(X) p Fm 1795 4607 a(j;l) p Fn 1906 4505 a(c) p Fm 1948 4520 a(j;l) p Fo 2022 4505 a(\() p Fn(H) p Fm 2141 4520 a(N) p Fk 2231 4505 a(\000) p Fn 2330 4505 a(E) p Fo 2408 4505 a(\)) t(~) p Fn 2446 4505 a(e) p Fm 2491 4520 a(j;l) p Fo 1011 4752 a(=) p Fi 1170 4669 a(X) p Fm 1195 4854 a(j;l) p Fn 1306 4752 a(c) p Fm 1348 4767 a(j;l) p Fn 1422 4752 a(t) p Fm 1457 4767 a(z) p Fe 1490 4777 a(j) p Fo 1527 4752 a(\() p Fn(H) p Fl 1646 4767 a(1) p Fk 1707 4752 a(\000) p Fn 1807 4752 a(E) p Fo 1885 4752 a(\)) p Fn(\021) p Fl 1971 4767 a(0) p Fn 2010 4752 a(e) p Fm 2055 4767 a(l) p Fo 2109 4752 a(=) p Fi 2213 4669 a(X) p Fm 2238 4854 a(j;l) p Fn 2349 4752 a(c) p Fm 2391 4767 a(j;l) p Fn 2465 4752 a(t) p Fm 2500 4767 a(z) p Fe 2533 4777 a(j) p Fo 2570 4752 a([) p Fn(H) p Fl 2678 4767 a(1) p Fn 2718 4752 a(;) p 2762 4752 a(\021) p Fl 2810 4767 a(0) p Fo 2849 4752 a(]) p Fn(e) p Fm 2921 4767 a(l) p Fn 2947 4752 a(:) p Fo 1902 5214 a(53) p eop %%Page: 54 54 54 53 bop Fo 0 631 a(The) p 193 631 a(v) m(ectors) p Fn 517 631 a(t) p Fm 552 646 a(z) p Fe 585 656 a(j) p Fo 621 631 a([) p Fn(H) p Fl 729 646 a(1) p Fn 769 631 a(;) p 813 631 a(\021) p Fl 861 646 a(0) p Fo 900 631 a(]) p Fn(e) p Fm 972 646 a(l) p Fo 1023 631 a(and) p Fn 1206 631 a(t) p Fm 1241 646 a(z) p Fe 1274 665 a(j) p Fb 1302 651 a(0) p Fo 1333 631 a([) p Fn(H) p Fl 1441 646 a(1) p Fn 1480 631 a(;) p 1524 631 a(\021) p Fl 1572 646 a(0) p Fo 1611 631 a(]) p Fn(e) p Fm 1683 646 a(l) p Fo 1735 631 a(are) p 1890 631 a(m) m(utually) p 2294 631 a(orthogonal) p 2775 631 a(if) p Fn 2857 631 a(j) p Fk 2930 631 a(6) p Fo(=) p Fn 3034 631 a(j) p Fg 3080 595 a(0) p Fo 3103 631 a(.) p 3171 631 a(Applying) p 3586 631 a(Lemma) 0 751 y(8.4) p 157 751 a(and) p 347 751 a(Lemma) p 695 751 a(8.6) p 852 751 a(to) p 971 751 a(the) p 1139 751 a(case) p Fn 1346 751 a(N) p Fo 1462 751 a(=) p 1566 751 a(1,) p Fn 1674 751 a( ) p Fm 1737 766 a(\017) p Fo 1798 751 a(=) p Fn 1901 751 a(e) p Fm 1946 766 a(l) p Fo 2005 751 a(and) p Fn 2195 751 a(\030) p Fm 2238 766 a(\017) p Fo 2298 751 a(=) p 2401 751 a(0,) p 2510 751 a(w) m(e) p 2653 751 a(ha) m(v) m(e) p Fk 1174 971 a(jj) p Fn(t) p Fm 1265 986 a(z) p Fe 1298 996 a(j) p Fo 1334 971 a([) p Fn(H) p Fl 1442 986 a(1) p Fn 1482 971 a(;) p 1526 971 a(\021) p Fl 1574 986 a(0) p Fo 1613 971 a(]) p Fn(e) p Fm 1685 986 a(l) p Fk 1711 971 a(jj) p Fl 1767 930 a(2) p Fk 1834 971 a(\024) p Fn 1939 971 a(C) p Fl 2009 986 a(9) p Fn 2048 971 a(e) p Fg 2093 930 a(\000) p Fm(cR) p Fd 2232 907 a(2) p Fn 2271 971 a(;) p Fo 2347 971 a(if) p Fn 2444 971 a(R) p Fk 2547 971 a(\025) p Fo 2652 971 a(1) p Fn(;) p Fo 0 1191 a(where) p Fn 288 1191 a(C) p Fl 358 1206 a(9) p Fo 437 1191 a(and) p Fn 633 1191 a(c) p Fo 714 1191 a(is) p 819 1191 a(a) p 906 1191 a(constan) m(t) p 1306 1191 a(dep) s(enden) m(t) p 1784 1191 a(only) p 2005 1191 a(on) p Fn 2147 1191 a(B) p Fo 2226 1191 a(,) p Fn 2294 1191 a(E) p Fo 2372 1191 a(.) p 2462 1191 a(By) p 2622 1191 a(this) p 2819 1191 a(inequalit) m(y) p 3278 1191 a(and) p 3474 1191 a(\(8.25\),) p 3791 1191 a(w) m(e) 0 1312 y(ha) m(v) m(e) p Fk 1250 1532 a(jj) p Fo(\() p Fn(H) p Fm 1425 1547 a(N) p Fk 1513 1532 a(\000) p Fn 1613 1532 a(E) p Fo 1691 1532 a(\)) t(~) p Fn 1729 1532 a(e) p Fk(jj) p Fl 1830 1491 a(2) p Fo 1090 1752 a(=) p Fm 1279 1644 a(N) p Fi 1251 1669 a(X) p Fm 1250 1851 a(j) p Fl 1283 1851 a(=1) p Fk 1389 1752 a(jj) p Fm 1501 1644 a(k) p Fi 1461 1669 a(X) p Fm 1465 1854 a(l) p Fl 1487 1854 a(=1) p Fn 1597 1752 a(c) p Fm 1639 1767 a(j;l) p Fn 1713 1752 a(t) p Fm 1748 1767 a(z) p Fe 1781 1777 a(j) p Fo 1818 1752 a([) p Fn(H) p Fl 1926 1767 a(1) p Fn 1965 1752 a(;) p 2009 1752 a(\021) p Fl 2057 1767 a(0) p Fo 2097 1752 a(]) p Fn(e) p Fm 2169 1767 a(l) p Fk 2195 1752 a(jj) p Fl 2251 1711 a(2) p Fk 1089 2068 a(\024) p Fm 1279 1960 a(N) p Fi 1251 1985 a(X) p Fm 1250 2168 a(j) p Fl 1283 2168 a(=1) p Fi 1389 1922 a( ) p Fm 1495 1960 a(k) p Fi 1454 1985 a(X) p Fm 1458 2170 a(l) p Fl 1480 2170 a(=1) p Fk 1591 2068 a(j) p Fn(c) p Fm 1661 2083 a(j;l) p Fk 1735 2068 a(j) p Fl 1763 2027 a(2) p Fi 1802 1922 a(!) p 1884 1922 a( ) p Fm 1990 1960 a(k) p Fi 1950 1985 a(X) p Fm 1954 2170 a(l) p Fl 1976 2170 a(=1) p Fk 2086 2068 a(jj) p Fo([) p Fn(H) p Fl 2250 2083 a(1) p Fn 2289 2068 a(;) p 2333 2068 a(\021) p Fl 2381 2083 a(0) p Fo 2420 2068 a(]) p Fn(e) p Fm 2492 2083 a(l) p Fk 2519 2068 a(jj) p Fl 2575 2027 a(2) p Fi 2613 1922 a(!) p Fk 1089 2384 a(\024) p Fn 1250 2384 a(k) s(C) p Fl 1374 2399 a(9) p Fn 1413 2384 a(e) p Fg 1458 2343 a(\000) p Fm(cR) p Fd 1597 2320 a(2) p Fm 1682 2276 a(N) p Fi 1654 2301 a(X) p Fm 1652 2484 a(j) p Fl 1685 2484 a(=1) p Fm 1832 2276 a(k) p Fi 1792 2301 a(X) p Fm 1795 2486 a(l) p Fl 1817 2486 a(=1) p Fk 1928 2384 a(j) p Fn(c) p Fm 1998 2399 a(j;l) p Fk 2072 2384 a(j) p Fl 2100 2343 a(2) p Fk 1089 2635 a(\024) p Fn 1250 2635 a(k) s(C) p Fl 1374 2650 a(9) p Fn 1413 2635 a(C) p Fm 1483 2650 a(k) p Fn 1525 2635 a(e) p Fg 1570 2594 a(\000) p Fm(cR) p Fd 1709 2570 a(2) p Fk 1748 2635 a(jj) p Fo 1808 2635 a(~) p Fn 1804 2635 a(e) p Fk(jj) p Fl 1905 2594 a(2) p Fo 1971 2635 a(=) p Fn 2075 2635 a(C) p 2152 2635 a(e) p Fg 2197 2594 a(\000) p Fm(cR) p Fd 2336 2570 a(2) p Fn 2375 2635 a(;) p Fo 2451 2635 a(if) p Fn 2547 2635 a(R) p Fk 2650 2635 a(\025) p Fn 2755 2635 a(R) p Fl 2829 2650 a(1) p Fn 2869 2635 a(;) p Fo 0 2855 a(where) p Fn 291 2855 a(C) p Fo 409 2855 a(is) p 516 2855 a(a) p 606 2855 a(constan) m(t) p 1009 2855 a(dep) s(enden) m(t) p 1489 2855 a(only) p 1712 2855 a(on) p Fn 1857 2855 a(B) p Fo 1936 2855 a(,) p Fn 2006 2855 a(E) p Fo 2084 2855 a(.) p 2182 2855 a(Since) p Fn 2445 2855 a(R) p Fk 2563 2855 a(\025) p Fn 2684 2855 a(R) p Fl 2758 2870 a(3) p Fo 2839 2855 a(and) p Fn 3037 2855 a(R) p Fl 3111 2870 a(3) p Fo 3193 2855 a(is) p 3300 2855 a(arbitrary) p 3675 2855 a(,) p 3744 2855 a(this) 0 2975 y(con) m(tradicts) p 502 2975 a(\(8.26\).) p 822 2975 a(Th) m(us) p 1069 2975 a(\(8.19\)) p 1351 2975 a(holds.) 146 3095 y(Next,) p 415 3095 a(w) m(e) p 561 3095 a(assume) p Fn 901 3095 a(N) p Fo 1024 3095 a(is) p 1125 3095 a(\014nite) p 1377 3095 a(\(when) p Fn 1672 3095 a(N) p Fo 1792 3095 a(=) p Fk 1900 3095 a(1) p Fo(,) p 2062 3095 a(the) p 2233 3095 a(inequalit) m(y) p 2688 3095 a(\(8.27\)) p 2972 3095 a(b) s(elo) m(w) p 3251 3095 a(is) p 3352 3095 a(ob) m(vious\)) p 3745 3095 a(and) 0 3216 y(pro) m(v) m(e) p 263 3216 a(there) p 512 3216 a(exists) p 781 3216 a(a) p 863 3216 a(constan) m(t) p Fn 1256 3216 a(R) p Fl 1330 3231 a(2) p Fn 1398 3216 a(>) p 1501 3216 a(R) p Fl 1575 3231 a(0) p Fo 1647 3216 a(dep) s(enden) m(t) p 2119 3216 a(only) p 2333 3216 a(on) p Fn 2468 3216 a(B) p Fo 2547 3216 a(,) p Fn 2607 3216 a(E) p Fo 2685 3216 a(,) p Fn 2745 3216 a(\016) p Fl 2788 3231 a(0) p Fo 2860 3216 a(and) p Fn 3050 3216 a(\013) p Fo 3145 3216 a(suc) m(h) p 3365 3216 a(that) 1008 3436 y(dim) p 1188 3436 a(Ran) p Fn 1379 3436 a(P) p Fl 1442 3451 a(\() p Fm(E) p Fg 1525 3451 a(\000) p Fm(\016) p Fd 1611 3460 a(0) p Fm 1646 3451 a(;E) p Fl 1722 3451 a(+) p Fm(\016) p Fd 1808 3460 a(0) p Fl 1841 3451 a(\)) p Fo 1873 3436 a(\() p Fn(H) p Fm 1992 3451 a(N) p Fo 2059 3436 a(\)) p Fk 2124 3436 a(\024) p Fn 2230 3436 a(N) p 2318 3436 a(k) s(;) p Fo 2448 3436 a(if) p Fn 2545 3436 a(R) p Fk 2648 3436 a(\025) p Fn 2753 3436 a(R) p Fl 2827 3451 a(2) p Fn 2866 3436 a(:) p Fo 3653 3436 a(\(8.27\)) 146 3656 y(W) p 238 3656 a(e) p 314 3656 a(prepare) p 667 3656 a(three) p 916 3656 a(lemmas.) p Fq 146 3859 a(Lemma) p 543 3859 a(8.7) p Ff 735 3859 a(L) p 791 3859 a(et) p Fn 913 3859 a(P) p Ff 1033 3859 a(b) p 1073 3859 a(e) p 1162 3859 a(an) p 1310 3859 a(ortho) p 1528 3859 a(gonal) p 1796 3859 a(pr) p 1882 3859 a(oje) p 2002 3859 a(ction) p 2257 3859 a(and) p Fn 2455 3859 a(P) p Fg 2532 3823 a(0) p Ff 2599 3859 a(b) p 2639 3859 a(e) p 2727 3859 a(a) p 2821 3859 a(\014nite) p 3082 3859 a(r) p 3118 3859 a(ank) p 3311 3859 a(op) p 3406 3859 a(er) p 3487 3859 a(ator) p 3704 3859 a(on) p 3852 3859 a(a) 0 3980 y(Hilb) p 167 3980 a(ert) p 320 3980 a(sp) p 405 3980 a(ac) p 495 3980 a(e.) p 614 3980 a(Supp) p 816 3980 a(ose) p 985 3980 a(that) p Fk 1643 4100 a(jj) p Fn(P) p Fk 1797 4100 a(\000) p Fn 1897 4100 a(P) p Fg 1974 4059 a(0) p Fk 1996 4100 a(jj) p Fn 2079 4100 a(<) p Fo 2183 4100 a(1) p Fn(:) p Ff 0 4274 a(Then,) p Fo 284 4274 a(dim) p 463 4274 a(Ran) p Fn 654 4274 a(P) p Fk 759 4274 a(\024) p Fo 864 4274 a(dim) p 1043 4274 a(Ran) p Fn 1234 4274 a(P) p Fg 1311 4238 a(0) p Ff 1334 4274 a(.) 146 4503 y(Pr) p 248 4503 a(o) p 293 4503 a(of.) p Fo 435 4503 a(See) p 609 4503 a([Co-Ne,) p 964 4503 a(Prop) s(osition) p 1489 4503 a(I) s(I) s(I.1].) p 1773 4435 64 4 v 1773 4528 4 93 v 1833 4528 V 1773 4531 64 4 v 1902 5214 a(54) p eop %%Page: 55 55 55 54 bop Fq 146 631 a(Lemma) p 543 631 a(8.8) p Ff 735 631 a(Ther) p 936 631 a(e) p 1016 631 a(exist) p 1242 631 a(smo) p 1407 631 a(oth) p 1574 631 a(functions) p Fk 1997 631 a(f) p Fn(\021) p Fm 2095 646 a(m) p Fk 2161 631 a(g) p Fm 2211 646 a(m) p Fg(2) p Fl(\000) p Ff 2369 631 a(,) p 2434 631 a(satisfying) p Fo 1349 821 a(0) p Fk 1425 821 a(\024) p Fn 1530 821 a(\021) p Fm 1578 836 a(m) p Fk 1673 821 a(\024) p Fo 1778 821 a(1) p 1862 821 a(for) p Fn 2013 821 a(m) p Fk 2126 821 a(2) p Fo 2220 821 a(\000) p Fn(;) p Fi 2542 738 a(X) p Fm 2526 921 a(m) p Fg(2) p Fl(\000) p Fn 2696 821 a(\021) p Fl 2748 780 a(2) p Fm 2744 845 a(m) p Fo 2838 821 a(=) p 2942 821 a(1) p 3025 821 a(on) p Fj 3163 821 a(R) p Fl 3250 779 a(2) p Fn 3290 821 a(;) p Fo 1430 1103 a(supp) p Fn 1648 1103 a(\021) p Fm 1696 1118 a(m) p Fk 1790 1103 a(\032) p Fn 1896 1103 a(K) p Fo 1986 1103 a(\() p Fn(m;) 2174 1035 y(R) p 2163 1079 98 4 v Fo 2163 1171 a(99) 2270 1103 y(\)) p 2474 1103 a(for) p Fn 2625 1103 a(m) p Fk 2738 1103 a(2) p Fo 2833 1103 a(\000) p Fl 2894 1118 a(0) p Fn 2933 1103 a(;) p Fo 1505 1340 a(supp) p Fn 1722 1340 a(\021) p Fm 1770 1355 a(m) p Fe 1832 1365 a(j) p Fk 1897 1340 a(\032) p 2002 1340 a(K) p Fm 2078 1355 a(j) p Fo 2115 1340 a(\() p Fn 2174 1272 a(R) p 2163 1316 V Fo 2163 1408 a(99) 2270 1340 y(\)) p 2474 1340 a(for) p Fn 2625 1340 a(m) p Fm 2710 1355 a(j) p Fk 2775 1340 a(2) p Fo 2869 1340 a(\000) p Fl 2930 1355 a(1) p Fn 2969 1340 a(;) p Fo 547 1577 a(sup) q(\() p Fk(jr) p Fn(\021) p Fm 891 1592 a(m) p Fk 958 1577 a(j) p Fl 986 1536 a(2) p Fo 1047 1577 a(+) p Fk 1145 1577 a(jr) p Fn(@) p Fm 1307 1592 a(x) p Fn 1351 1577 a(\021) p Fm 1399 1592 a(m) p Fk 1466 1577 a(j) p Fl 1494 1536 a(2) p Fo 1555 1577 a(+) p Fk 1653 1577 a(jr) p Fn(@) p Fm 1815 1592 a(x) p Fn 1859 1577 a(\021) p Fm 1907 1592 a(m) p Fk 1974 1577 a(j) p Fl 2002 1536 a(2) p Fk 2068 1577 a(\024) p Fn 2202 1509 a(C) p 2184 1553 115 4 v 2184 1645 a(R) p Fl 2259 1616 a(2) p Fo 2474 1577 a(for) p Fn 2625 1577 a(m) p Fk 2738 1577 a(2) p Fo 2833 1577 a(\000) p Fn(;) p Fo 2972 1577 a(if) p Fn 3071 1577 a(R) p Fk 3174 1577 a(\025) p Fo 3279 1577 a(1) p Fn(;) p Fo 3653 1577 a(\(8.28\)) p Ff 0 1792 a(wher) p 196 1792 a(e) p Fn 275 1792 a(C) p Ff 387 1792 a(is) p 492 1792 a(a) p 576 1792 a(c) p 616 1792 a(onstant) p 965 1792 a(indep) p 1190 1792 a(endent) p 1505 1792 a(of) p Fn 1620 1792 a(m) p Ff(,) p Fn 1770 1792 a(R) p Ff 1845 1792 a(.) 146 1986 y(Pr) p 248 1986 a(o) p 293 1986 a(of.) p Fo 435 1986 a(T) p 497 1986 a(ak) m(e) p Fn 671 1986 a(\020) p Fl 714 2001 a(0) p Fk 781 1986 a(2) p Fn 875 1986 a(C) p Fg 952 1949 a(1) p Fl 945 2010 a(0) p Fo 1027 1986 a(\() p Fn(K) p Fo 1155 1986 a(\(0) p Fn(;) p Fl 1313 1946 a(1) p 1295 1962 71 4 v 1295 2020 a(99) p Fo 1376 1986 a(\)\)) p 1484 1986 a(and) p Fn 1674 1986 a(\020) p Fl 1717 2001 a(1) p Fk 1783 1986 a(2) p Fn 1878 1986 a(C) p Fg 1955 1949 a(1) p Fl 1948 2010 a(0) p Fo 2029 1986 a(\() p Fk(K) p Fl 2143 2001 a(0) p Fo 2183 1986 a(\() p Fl 2248 1946 a(1) p 2231 1962 V 2231 2020 a(99) p Fo 2311 1986 a(\)\)) p 2419 1986 a(satisfying) 1041 2229 y(0) p Fk 1117 2229 a(\024) p Fn 1222 2229 a(\020) p Fl 1265 2244 a(0) p Fk 1332 2229 a(\024) p Fo 1437 2229 a(1) p 1519 2229 a(on) p Fj 1654 2229 a(R) p Fl 1741 2187 a(2) p Fn 1781 2229 a(;) p 2006 2229 a(\020) p Fl 2049 2244 a(0) p Fo 2116 2229 a(=) p 2220 2229 a(1) p 2301 2229 a(on) p Fn 2436 2229 a(K) p Fo 2526 2229 a(\(0) p Fn(;) p Fo 2716 2162 a(1) p 2667 2206 147 4 v 2667 2298 a(100) 2823 2229 y(\)) 1041 2462 y(0) p Fk 1117 2462 a(\024) p Fn 1222 2462 a(\020) p Fl 1265 2477 a(1) p Fk 1332 2462 a(\024) p Fo 1437 2462 a(1) p 1519 2462 a(on) p Fj 1654 2462 a(R) p Fl 1741 2420 a(2) p Fn 1781 2462 a(;) p 2006 2462 a(\020) p Fl 2049 2477 a(1) p Fo 2116 2462 a(=) p 2220 2462 a(1) p 2301 2462 a(on) p Fk 2436 2462 a(K) p Fl 2512 2477 a(0) p Fo 2552 2462 a(\() 2649 2395 y(1) p 2600 2439 V 2600 2531 a(100) 2756 2462 y(\)) 0 2678 y(and) p 190 2678 a(put) p Fn 661 2925 a(\021) p Fm 709 2940 a(m) p Fo 859 2925 a(=) p Fn 1708 2854 a(\020) p Fl 1751 2869 a(0) p Fo 1790 2854 a(\() p Fm 1838 2815 a(z) p Fg 1874 2815 a(\000) p 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a(j) p Fo 859 3308 a(=) p Fn 1692 3237 a(\020) p Fl 1735 3252 a(1) p Fo 1774 3237 a(\() p Fm 1822 3187 a(z) p Fg 1858 3187 a(\000) p Fm(m) p Fe 1975 3197 a(j) p 1822 3214 186 4 v Fm 1888 3272 a(R) p Fo 2017 3237 a(\)) p 1028 3285 1692 4 v Fi 1028 3301 a(r) p 1111 3301 1609 4 v 1111 3360 a(P) p Fm 1198 3447 a(m) p Fb 1260 3428 a(0) p Fg 1283 3447 a(2) p Fl(\000) p Fd 1374 3456 a(0) p Fn 1430 3426 a(\020) p Fl 1473 3441 a(0) p Fo 1512 3426 a(\() p Fm 1560 3387 a(z) p Fg 1596 3387 a(\000) p Fm(m) p Fb 1713 3368 a(0) p 1560 3403 176 4 v Fm 1620 3460 a(R) p Fo 1745 3426 a(\)) p Fl 1783 3397 a(2) p Fo 1844 3426 a(+) p Fi 1942 3360 a(P) p Fm 2030 3447 a(m) p Fe 2092 3466 a(j) p Fb 2120 3452 a(0) p Fg 2147 3447 a(2) p Fl(\000) p Fd 2238 3456 a(1) p Fn 2294 3426 a(\020) p Fl 2337 3441 a(1) p Fo 2376 3426 a(\() p Fm 2424 3367 a(z) p Fg 2460 3367 a(\000) p Fm(m) p Fe 2577 3386 a(j) p Fb 2605 3372 a(0) p 2424 3403 208 4 v Fm 2501 3460 a(R) p Fo 2642 3426 a(\)) p Fl 2680 3397 a(2) p Fo 2826 3308 a(\() p Fn(m) p Fm 2949 3323 a(j) p Fk 3014 3308 a(2) p Fo 3108 3308 a(\000) p Fl 3169 3323 a(1) p Fo 3208 3308 a(\)) p Fn(:) p Fo 0 3646 a(One) p 206 3646 a(can) p 385 3646 a(easily) p 654 3646 a(c) m(hec) m(k) p 916 3646 a(that) p 1128 3646 a(the) p 1296 3646 a(functions) p 1716 3646 a(ab) s(o) m(v) m(e) p 1993 3646 a(ha) m(v) m(e) p 2218 3646 a(desired) p 2549 3646 a(prop) s(erties.) p 3045 3578 64 4 v 3045 3671 4 93 v 3105 3671 V 3045 3674 64 4 v Fq 146 3857 a(Lemma) p 543 3857 a(8.9) p Ff 735 3857 a(F) p 792 3857 a(or) p Fn 917 3857 a(m) p Fk 1030 3857 a(2) p Fo 1124 3857 a(\000) p Ff(,) p 1250 3857 a(put) p Fn 1420 3857 a(\021) p Fl 1472 3806 a(\() p Fm(m) p Fl(\)) 1468 3878 y(0) p Fo 1593 3857 a(\() p Fn(z) p Fo 1680 3857 a(\)) p 1746 3857 a(=) p Fn 1850 3857 a(\021) p Fm 1898 3872 a(m) p Fo 1964 3857 a(\() p Fn(z) p Fo 2074 3857 a(+) p Fn 2172 3857 a(m) p Fo(\)) p Ff(.) p 2370 3857 a(De\014ne) p 2678 3857 a(the) p 2840 3857 a(op) p 2935 3857 a(er) p 3016 3857 a(ator) p Fn 3223 3857 a(A) p Fm 3296 3872 a(N) p Fo 3364 3857 a(\() p Fn(w) p Fo 3475 3857 a(\)) p Ff 3547 3857 a(by) p Fn 394 4063 a(A) p Fm 467 4078 a(N) p Fo 534 4063 a(\() p Fn(w) p Fo 645 4063 a(\)) p 710 4063 a(=) p Fi 864 3980 a(X) p Fm 813 4164 a(m) p Fe 875 4174 a(j) p Fg 908 4164 a(2) p Fl(\000) p Fd 999 4173 a(1) p Fn 1051 4063 a(t) p Fm 1086 4078 a(z) p Fe 1119 4088 a(j) p Fn 1155 4063 a(\021) p Fl 1207 4009 a(\() p Fm(m) p Fe 1296 4019 a(j) p Fl 1330 4009 a(\)) 1203 4085 y(0) p Fo 1361 4063 a(\() p Fn(H) p Fl 1480 4078 a(1) p Fk 1541 4063 a(\000) p Fn 1641 4063 a(w) p Fo 1714 4063 a(\)) p Fg 1752 4022 a(\000) p Fl(1) p Fn 1846 4063 a(t) p Fg 1881 4078 a(\000) p Fm(z) p Fe 1969 4088 a(j) p Fn 2005 4063 a(\021) p Fm 2053 4078 a(m) p Fe 2115 4088 a(j) p Fo 2175 4063 a(+) p Fi 2307 3980 a(X) p Fm 2273 4164 a(m) p Fg(2) p Fl(\000) p Fd 2426 4173 a(0) p Fn 2477 4063 a(t) p Fm 2512 4078 a(m) p Fn 2579 4063 a(\021) p Fl 2631 4012 a(\() p Fm(m) p Fl(\)) 2627 4085 y(0) p Fo 2752 4063 a(\() p Fn(H) p Fl 2871 4078 a(0) p Fk 2933 4063 a(\000) p Fn 3032 4063 a(w) p Fo 3105 4063 a(\)) p Fg 3143 4022 a(\000) p Fl(1) p Fn 3237 4063 a(t) p Fg 3272 4078 a(\000) p Fm(m) p Fn 3394 4063 a(\021) p Fm 3442 4078 a(m) p Ff 0 4344 a(for) p Fn 168 4344 a(w) p Fk 290 4344 a(2) p Fj 407 4344 a(C) p Ff 495 4344 a(,) p Fk 575 4344 a(j) p Fn(w) p Fk 707 4344 a(\000) p Fn 815 4344 a(E) p Fk 893 4344 a(j) p Fo 971 4344 a(=) p Fn 1097 4344 a(\016) p Fl 1140 4359 a(0) p Ff 1180 4344 a(.) p 1291 4344 a(Then,) p 1590 4344 a(the) p 1764 4344 a(sums) p 2023 4344 a(c) p 2063 4344 a(onver) p 2294 4344 a(ge) p 2430 4344 a(in) p 2562 4344 a(the) p 2736 4344 a(str) p 2844 4344 a(ong) p 3041 4344 a(op) p 3136 4344 a(er) p 3217 4344 a(ator) p 3437 4344 a(top) p 3564 4344 a(olo) p 3684 4344 a(gy) p 3822 4344 a(of) 0 4465 y(the) p 170 4465 a(b) p 210 4465 a(ounde) p 457 4465 a(d) p 549 4465 a(op) p 644 4465 a(er) p 725 4465 a(ator) p 941 4465 a(fr) p 1007 4465 a(om) p Fn 1180 4465 a(L) p Fl 1246 4429 a(2) p Fo 1285 4465 a(\() p Fj(R) p Fl 1411 4423 a(2) p Fo 1450 4465 a(\)) p Ff 1531 4465 a(to) p Fn 1656 4465 a(D) p Fo 1740 4465 a(\() p Fn(H) p Fm 1859 4480 a(N) p Fo 1926 4465 a(\)) p Ff(.) p 2063 4465 a(Mor) p 2236 4465 a(e) p 2276 4465 a(over,) p 2532 4465 a(ther) p 2695 4465 a(e) p 2783 4465 a(exists) p 3058 4465 a(a) p 3150 4465 a(c) p 3190 4465 a(onstant) p Fn 3547 4465 a(C) p Fl 3617 4480 a(10) p Fn 3735 4465 a(>) p Fo 3853 4465 a(0) p Ff 0 4585 a(dep) p 140 4585 a(endent) p 456 4585 a(on) p Fn 595 4585 a(B) p Ff 674 4585 a(,) p Fn 739 4585 a(E) p Ff 817 4585 a(,) p Fn 882 4585 a(\016) p Fl 925 4600 a(0) p Ff 965 4585 a(,) p 1029 4585 a(and) p Fn 1219 4585 a(\013) p Ff 1316 4585 a(such) p 1538 4585 a(that) p Fk 1313 4775 a(jj) p Fn(A) p Fm 1442 4790 a(N) p Fo 1509 4775 a(\() p Fn(w) p Fo 1620 4775 a(\)) p Fk(jj) p Fm 1714 4792 a(B) p Fl 1770 4792 a(\() p Fm(L) p Fd 1845 4773 a(2) p Fl 1879 4792 a(\() p Fa(R) p Fd 1967 4773 a(2) p Fl 2002 4792 a(\);) p Fm(D) p Fl 2109 4792 a(\() p Fm(H) p Fe 2194 4803 a(N) p Fl 2252 4792 a(\)\)) p Fk 2339 4775 a(\024) p Fn 2444 4775 a(C) p Fl 2514 4790 a(10) p Ff 0 4965 a(for) p 156 4965 a(every) p Fn 413 4965 a(w) p Fk 514 4965 a(2) p Fj 608 4965 a(C) p Ff 731 4965 a(with) p Fk 943 4965 a(j) p Fn(w) p Fk 1065 4965 a(\000) p Fn 1164 4965 a(E) p Fk 1242 4965 a(j) p Fo 1298 4965 a(=) p Fn 1401 4965 a(\016) p Fl 1444 4980 a(0) p Ff 1484 4965 a(,) p 1549 4965 a(if) p Fn 1643 4965 a(R) p Fk 1746 4965 a(\025) p Fo 1851 4965 a(1) p Ff(.) p Fo 1902 5214 a(55) p eop %%Page: 56 56 56 55 bop Ff 146 631 a(Pr) p 248 631 a(o) p 293 631 a(of.) p Fo 440 631 a(It) p 550 631 a(is) p 652 631 a(su\016cien) m(t) p 1066 631 a(to) p 1189 631 a(sho) m(w) p 1435 631 a(that) p 1651 631 a(there) p 1905 631 a(exists) p 2178 631 a(a) p 2264 631 a(constan) m(t) p Fn 2661 631 a(C) p Fl 2731 646 a(10) p Fn 2841 631 a(>) p Fo 2952 631 a(0) p 3037 631 a(dep) s(enden) m(t) p 3513 631 a(on) p Fn 3653 631 a(B) p Fo 3732 631 a(,) p Fn 3797 631 a(E) p Fo 3875 631 a(,) p Fn 0 751 a(\016) p Fl 43 766 a(0) p Fo 83 751 a(,) p 142 751 a(and) p Fn 332 751 a(\013) p Fo 427 751 a(suc) m(h) p 647 751 a(that) p Fk 1181 872 a(jj) p Fn(A) p Fm 1310 887 a(N) p Fo 1376 872 a(\() p Fn(w) p Fo 1487 872 a(\)) p Fn(\036) p Fk(jj) p Fl 1639 831 a(2) p Fm 1639 896 a(B) p Fl 1695 896 a(\() p Fm(L) p Fd 1770 877 a(2) p Fl 1804 896 a(\() p Fa(R) p Fd 1892 877 a(2) p Fl 1927 896 a(\);) p Fm(D) p Fl 2034 896 a(\() p Fm(H) p Fe 2119 907 a(N) p Fl 2177 896 a(\)\)) p Fk 2264 872 a(\024) p Fn 2369 872 a(C) p Fl 2439 887 a(10) p Fk 2513 872 a(jj) p Fn(\036) p Fk(jj) p Fl 2683 831 a(2) p Fo 3653 872 a(\(8.29\)) 0 1042 y(for) p 156 1042 a(ev) m(ery) p Fn 421 1042 a(\036) p Fk 519 1042 a(2) p Fn 625 1042 a(C) p Fg 702 1006 a(1) p Fl 695 1067 a(0) p Fo 777 1042 a(\() p Fj(R) p Fl 902 1000 a(2) p Fo 942 1042 a(\)) p 1019 1042 a(\(notice) p 1352 1042 a(that) p 1571 1042 a(the) p 1746 1042 a(sum) p 1960 1042 a(in) p 2081 1042 a(the) p 2256 1042 a(de\014nition) p 2697 1042 a(of) p Fn 2815 1042 a(A) p Fm 2888 1057 a(N) p Fo 2956 1042 a(\() p Fn(w) p Fo 3067 1042 a(\)) p Fn(\036) p Fo 3201 1042 a(b) s(ecomes) p 3597 1042 a(a) p 3685 1042 a(\014nite) 0 1163 y(sum) p 206 1163 a(for) p Fn 356 1163 a(\036) p Fk 441 1163 a(2) p Fn 535 1163 a(C) p Fg 612 1126 a(1) p Fl 605 1187 a(0) p Fo 687 1163 a(\() p Fj(R) p Fl 812 1121 a(2) p Fo 851 1163 a(\);) p 949 1163 a(the) p 1117 1163 a(strong) p 1416 1163 a(con) m(v) m(ergence) p 1955 1163 a(follo) m(ws) p 2275 1163 a(from) p 2505 1163 a(\(8.29\)) p 2787 1163 a(and) p 2977 1163 a(an) p Fn 3112 1163 a(\017=) p Fo(3-argumen) m(t\).) 146 1300 y(T) p 208 1300 a(ak) m(e) p Fn 390 1300 a(\036) p Fk 489 1300 a(2) p Fn 596 1300 a(C) p Fg 673 1264 a(1) p Fl 666 1324 a(0) p Fo 748 1300 a(\() p Fj(R) p Fl 873 1258 a(2) p Fo 913 1300 a(\).) p 1044 1300 a(Then) p 1307 1300 a(w) m(e) p 1458 1300 a(ha) m(v) m(e) p 1691 1300 a(\() p Fn(H) p Fl 1810 1315 a(1) p Fk 1877 1300 a(\000) p Fn 1982 1300 a(w) p Fo 2055 1300 a(\)) p Fg 2093 1264 a(\000) p Fl(1) p Fn 2187 1300 a(t) p Fg 2222 1315 a(\000) p Fm(z) p Fe 2310 1325 a(j) p Fn 2346 1300 a(\021) p Fm 2394 1315 a(m) p Fe 2456 1325 a(j) p Fn 2493 1300 a(\036) p Fk 2592 1300 a(2) p Fn 2700 1300 a(D) p Fo 2784 1300 a(\() p Fn(H) p Fl 2903 1315 a(1) p Fo 2942 1300 a(\).) p 3073 1300 a(Th) m(us) p Fn 3328 1300 a(t) p Fm 3363 1315 a(z) p Fe 3396 1325 a(j) p Fn 3433 1300 a(\021) p Fl 3485 1246 a(\() p Fm(m) p Fe 3574 1256 a(j) p Fl 3607 1246 a(\)) 3481 1322 y(0) p Fo 3639 1300 a(\() p Fn(H) p Fl 3758 1315 a(1) p Fk 3825 1300 a(\000) p Fn 0 1420 a(w) p Fo 73 1420 a(\)) p Fg 111 1384 a(\000) p Fl(1) p Fn 205 1420 a(t) p Fg 240 1435 a(\000) p Fm(z) p Fe 328 1445 a(j) p Fn 364 1420 a(\021) p Fm 412 1435 a(m) p Fe 474 1445 a(j) p Fn 512 1420 a(\036) p Fk 597 1420 a(2) p Fn 691 1420 a(D) p Fo 775 1420 a(\() p Fn(H) p Fm 894 1435 a(N) p Fo 961 1420 a(\)) p 1027 1420 a(b) m(y) p 1158 1420 a(Lemma) p 1502 1420 a(8.2,) p 1683 1420 a(for) p Fn 1828 1420 a(m) p Fm 1913 1435 a(j) p Fk 1977 1420 a(2) p Fo 2071 1420 a(\000) p Fl 2132 1435 a(1) p Fo 2172 1420 a(.) p 2241 1420 a(Similarly) p 2651 1420 a(w) m(e) p 2791 1420 a(ha) m(v) m(e) p 3011 1420 a(\() p Fn(H) p Fl 3130 1435 a(0) p Fk 3183 1420 a(\000) p Fn 3274 1420 a(w) p Fo 3347 1420 a(\)) p Fg 3385 1384 a(\000) p Fl(1) p Fn 3479 1420 a(t) p Fg 3514 1435 a(\000) p Fm(m) p Fn 3635 1420 a(\021) p Fm 3683 1435 a(m) p Fn 3750 1420 a(\036) p Fk 3836 1420 a(2) p Fn 0 1564 a(D) p Fo 84 1564 a(\() p Fn(H) p Fl 203 1579 a(0) p Fo 242 1564 a(\)) p 312 1564 a(and) p Fn 502 1564 a(t) p Fm 537 1579 a(m) p Fn 604 1564 a(\021) p Fl 656 1513 a(\() p Fm(m) p Fl(\)) 652 1586 y(0) p Fo 777 1564 a(\() p Fn(H) p Fl 896 1579 a(0) p Fk 957 1564 a(\000) p Fn 1057 1564 a(w) p Fo 1130 1564 a(\)) p Fg 1168 1528 a(\000) p Fl(1) p Fn 1262 1564 a(t) p Fg 1297 1579 a(\000) p Fm(m) p Fn 1418 1564 a(\021) p Fm 1466 1579 a(m) p Fn 1533 1564 a(\036) p Fk 1618 1564 a(2) p Fn 1713 1564 a(D) p Fo 1797 1564 a(\() p Fn(H) p Fm 1916 1579 a(N) p Fo 1982 1564 a(\)) p 2053 1564 a(for) p Fn 2202 1564 a(m) p Fk 2315 1564 a(2) p Fo 2409 1564 a(\000) p Fl 2470 1579 a(0) p Fo 2509 1564 a(,) p 2569 1564 a(b) m(y) p 2704 1564 a(Lemma) p 3052 1564 a(8.3.) 146 1684 y(Since) p 401 1684 a(the) p 569 1684 a(supp) s(ort) p 930 1684 a(of) p 1041 1684 a(the) p 1209 1684 a(functions) p Fk 1209 1874 a(f) p Fn(t) p Fm 1294 1889 a(z) p Fe 1327 1899 a(j) p Fn 1364 1874 a(\021) p Fl 1416 1820 a(\() p Fm(m) p Fe 1505 1830 a(j) p Fl 1538 1820 a(\)) 1412 1896 y(0) p Fo 1569 1874 a(\() p Fn(H) p Fl 1688 1889 a(1) p Fk 1750 1874 a(\000) p Fn 1849 1874 a(w) p Fo 1922 1874 a(\)) p Fg 1960 1833 a(\000) p Fl(1) p Fn 2054 1874 a(t) p Fg 2089 1889 a(\000) p Fm(z) p Fe 2177 1899 a(j) p Fn 2213 1874 a(\021) p Fm 2261 1889 a(m) p Fe 2323 1899 a(j) p Fn 2361 1874 a(\036) p Fk(g) p Fm 2469 1889 a(m) p Fe 2531 1899 a(j) p Fg 2563 1889 a(2) p Fl(\000) p Fd 2654 1898 a(1) p Fo 0 2052 a(are) p 166 2052 a(disjoin) m(t,) p 546 2052 a(w) m(e) p 693 2052 a(see) p 854 2052 a(that) p 1070 2052 a(these) p 1323 2052 a(functions) p 1748 2052 a(are) p 1914 2052 a(m) m(utually) p 2330 2052 a(orthogonal) p 2821 2052 a(in) p Fn 2939 2052 a(D) p Fo 3023 2052 a(\() p Fn(H) p Fm 3142 2067 a(N) p Fo 3208 2052 a(\).) p 3328 2052 a(Similarly) p 3706 2052 a(,) p 3767 2052 a(the) 0 2173 y(functions) p Fk 1158 2351 a(f) p Fn(t) p Fm 1243 2366 a(m) p Fn 1310 2351 a(\021) p Fl 1362 2300 a(\() p Fm(m) p Fl(\)) 1358 2373 y(0) p Fo 1483 2351 a(\() p Fn(H) p Fl 1602 2366 a(0) p Fk 1663 2351 a(\000) p Fn 1763 2351 a(w) p Fo 1836 2351 a(\)) p Fg 1874 2310 a(\000) p Fl(1) p Fn 1967 2351 a(t) p Fg 2002 2366 a(\000) p Fm(m) p Fn 2124 2351 a(\021) p Fm 2172 2366 a(m) p Fn 2239 2351 a(\036) p Fk(g) p Fm 2347 2366 a(m) p Fg(2) p Fl(\000) p Fd 2500 2378 a(0) p Fe(;\014) s(;\015) p Fn 2649 2351 a(;) p Fo 1043 2505 a(\000) p Fl 1104 2520 a(0) p Fm(;\014) s(;\015) p Fo 1293 2505 a(=) p 1397 2505 a(\000) p Fl 1458 2520 a(0) p Fk 1519 2505 a(\\) p 1608 2505 a(f) p Fn(m) p Fo(\() p Fn(p) p Fo(\);) p Fn 1944 2505 a(p) p Fk 2021 2505 a(2) p Fo 2115 2505 a(2) p Fj(Z) p Fl 2248 2463 a(2) p Fo 2309 2505 a(+) p 2407 2505 a(\() p Fn(\014) p 2506 2505 a(;) p 2550 2505 a(\015) p Fo 2606 2505 a(\)) p Fk(g) p Fo 0 2684 a(are) p 163 2684 a(m) m(utually) p 574 2684 a(orthogonal) p 1062 2684 a(in) p Fn 1176 2684 a(D) p Fo 1260 2684 a(\() p Fn(H) p Fm 1379 2699 a(N) p Fo 1446 2684 a(\),) p 1543 2684 a(for) p 1692 2684 a(eac) m(h) p Fn 1912 2684 a(\014) p Fo 2000 2684 a(=) p 2104 2684 a(0) p Fn(;) p Fo 2197 2684 a(1) p 2277 2684 a(and) p Fn 2467 2684 a(\015) p Fo 2551 2684 a(=) p 2654 2684 a(0) p Fn(;) p Fo 2747 2684 a(1.) 146 2804 y(W) p 238 2804 a(e) p 314 2804 a(ha) m(v) m(e) p Fk 838 2982 a(jj) p Fn(t) p Fm 929 2997 a(z) p Fe 962 3007 a(j) p Fn 998 2982 a(\021) p Fl 1050 2928 a(\() p Fm(m) p Fe 1139 2938 a(j) p Fl 1172 2928 a(\)) 1046 3004 y(0) p Fo 1203 2982 a(\() p Fn(H) p Fl 1322 2997 a(1) p Fk 1384 2982 a(\000) p Fn 1483 2982 a(w) p Fo 1556 2982 a(\)) p Fg 1594 2941 a(\000) p Fl(1) p Fn 1688 2982 a(t) p Fg 1723 2997 a(\000) p Fm(z) p Fe 1811 3007 a(j) p Fn 1848 2982 a(\021) p Fm 1896 2997 a(m) p Fe 1958 3007 a(j) p Fn 1995 2982 a(\036) p Fk(jj) p Fl 2109 2941 a(2) p Fm 2109 3007 a(N) p Fo 678 3154 a(=) p Fk 838 3154 a(jj) p Fn(\021) p Fl 946 3100 a(\() p Fm(m) p Fe 1035 3110 a(j) p Fl 1067 3100 a(\)) 942 3176 y(0) p Fo 1099 3154 a(\() p Fn(H) p Fl 1218 3169 a(1) p Fk 1279 3154 a(\000) p Fn 1379 3154 a(w) p Fo 1452 3154 a(\)) p Fg 1490 3113 a(\000) p Fl(1) p Fn 1583 3154 a(t) p Fg 1618 3169 a(\000) p Fm(z) p Fe 1706 3179 a(j) p Fn 1743 3154 a(\021) p Fm 1791 3169 a(m) p Fe 1853 3179 a(j) p Fn 1890 3154 a(\036) p Fk(jj) p Fl 2004 3113 a(2) 2004 3179 y(1) p Fk 677 3300 a(\024) p Fn 838 3300 a(C) p Fl 908 3315 a(1) p Fk 947 3300 a(jj) p Fo(\() p Fn(H) p Fl 1122 3315 a(1) p Fk 1182 3300 a(\000) p Fn 1282 3300 a(w) p Fo 1355 3300 a(\)) p Fg 1393 3259 a(\000) p Fl(1) p Fn 1487 3300 a(t) p Fg 1522 3315 a(\000) p Fm(z) p Fe 1610 3325 a(j) p Fn 1646 3300 a(\021) p Fm 1694 3315 a(m) p Fe 1756 3325 a(j) p Fn 1794 3300 a(\036) p Fk(jj) p Fl 1908 3259 a(2) 1908 3324 y(1) p Fk 677 3445 a(\024) p Fn 838 3445 a(C) p Fl 908 3460 a(1) p Fo 947 3445 a(\(1) p 1056 3445 a(+) p Fn 1154 3445 a(E) p Fo 1254 3445 a(+) p Fn 1352 3445 a(\016) p Fl 1395 3460 a(0) p Fo 1435 3445 a(\)) p Fl 1473 3404 a(2) p Fo 1512 3445 a(\(1) p 1621 3445 a(+) p Fk 1719 3445 a(jj) p Fo(\() p Fn(H) p Fl 1894 3460 a(1) p Fk 1954 3445 a(\000) p Fn 2054 3445 a(w) p Fo 2127 3445 a(\)) p Fg 2165 3404 a(\000) p Fl(1) p Fk 2259 3445 a(jj) p Fl 2315 3404 a(2) p Fm 2315 3470 a(B) p Fl 2371 3470 a(\() p Fm(L) p Fd 2446 3451 a(2) p Fl 2481 3470 a(\() p Fa(R) p Fd 2569 3451 a(2) p Fl 2603 3470 a(\);) p Fm(L) p Fd 2698 3451 a(2) p Fl 2733 3470 a(\() p Fa(R) p Fd 2821 3451 a(2) p Fl 2856 3470 a(\)\)) p Fo 2915 3445 a(\)) p Fk(jj) p Fn(\021) p Fm 3057 3460 a(m) p Fe 3119 3470 a(j) p Fn 3155 3445 a(\036) p Fk(jj) p Fl 3269 3404 a(2) p Fo 0 3623 a(for) p Fn 146 3623 a(m) p Fm 231 3638 a(j) p Fk 296 3623 a(2) p Fo 390 3623 a(\000) p Fl 451 3638 a(1) p Fo 520 3623 a(and) p Fn 707 3623 a(w) p Fk 807 3623 a(2) p Fj 901 3623 a(C) p Fo 1019 3623 a(with) p Fk 1239 3623 a(j) p Fn(w) p Fk 1355 3623 a(\000) p Fn 1448 3623 a(E) p Fk 1526 3623 a(j) p Fo 1583 3623 a(=) p Fn 1686 3623 a(\016) p Fl 1729 3638 a(0) p Fo 1769 3623 a(,) p 1826 3623 a(if) p Fn 1913 3623 a(R) p Fk 2016 3623 a(\025) p Fo 2121 3623 a(1.) p 2239 3623 a(W) p 2331 3623 a(e) p 2404 3623 a(used) p 2624 3623 a(\(4.6\).) p 2894 3623 a(Notice) p 3198 3623 a(that) p 3406 3623 a(w) m(e) p 3547 3623 a(can) p 3723 3623 a(tak) m(e) 0 3744 y(the) p 168 3744 a(constan) m(t) p Fn 561 3744 a(C) p Fl 631 3759 a(1) p Fo 703 3744 a(in) p 817 3744 a(Lemma) p 1165 3744 a(4.3) p 1322 3744 a(uniformly) p 1767 3744 a(in) p Fn 1881 3744 a(R) p Fk 1983 3744 a(\025) p Fo 2088 3744 a(1,) p 2197 3744 a(b) m(y) p 2332 3744 a(\(8.28\).) p 2652 3744 a(Similarly) p 3030 3744 a(,) p 3086 3744 a(w) m(e) p 3229 3744 a(ha) m(v) m(e) p Fk 854 3936 a(jj) p Fn(t) p Fm 945 3951 a(m) p Fn 1011 3936 a(\021) p Fl 1063 3885 a(\() p Fm(m) p Fl(\)) 1059 3957 y(0) p Fo 1184 3936 a(\() p Fn(H) p Fl 1303 3951 a(0) p Fk 1364 3936 a(\000) p Fn 1464 3936 a(w) p Fo 1537 3936 a(\)) p Fg 1575 3894 a(\000) p Fl(1) p Fn 1669 3936 a(t) p Fg 1704 3951 a(\000) p Fm(m) p Fn 1825 3936 a(\021) p Fm 1873 3951 a(m) p Fn 1940 3936 a(\036) p Fk(jj) p Fl 2054 3894 a(2) p Fm 2054 3960 a(N) p Fk 693 4081 a(\024) p Fn 854 4081 a(C) p Fl 924 4096 a(1) p Fo 963 4081 a(\(1) p 1072 4081 a(+) p Fn 1170 4081 a(E) p Fo 1270 4081 a(+) p Fn 1368 4081 a(\016) p Fl 1411 4096 a(0) p Fo 1451 4081 a(\)) p Fl 1489 4040 a(2) p Fo 1528 4081 a(\(1) p 1637 4081 a(+) p Fk 1735 4081 a(jj) p Fo(\() p Fn(H) p Fl 1910 4096 a(0) p Fk 1971 4081 a(\000) p Fn 2070 4081 a(w) p Fo 2143 4081 a(\)) p Fg 2181 4040 a(\000) p Fl(1) p Fk 2275 4081 a(jj) p Fl 2331 4040 a(2) p Fm 2331 4106 a(B) p Fl 2387 4106 a(\() p Fm(L) p Fd 2462 4087 a(2) p Fl 2497 4106 a(\() p Fa(R) p Fd 2585 4087 a(2) p Fl 2620 4106 a(\);) p Fm(L) p Fd 2715 4087 a(2) p Fl 2749 4106 a(\() p Fa(R) p Fd 2837 4087 a(2) p Fl 2872 4106 a(\)\)) p Fo 2931 4081 a(\)) p Fk(jj) p Fn(\021) p Fm 3073 4096 a(m) p Fn 3139 4081 a(\036) p Fk(jj) p Fl 3253 4040 a(2) p Fo 0 4259 a(for) p Fn 149 4259 a(m) p Fk 262 4259 a(2) p Fo 356 4259 a(\000) p Fl 417 4274 a(0) p Fo 489 4259 a(and) p Fn 679 4259 a(w) p Fk 779 4259 a(2) p Fj 873 4259 a(C) p Fo 994 4259 a(with) p Fk 1216 4259 a(j) p Fn(w) p Fk 1338 4259 a(\000) p Fn 1438 4259 a(E) p Fk 1516 4259 a(j) p Fo 1571 4259 a(=) p Fn 1675 4259 a(\016) p Fl 1718 4274 a(0) p Fo 1757 4259 a(,) p 1817 4259 a(if) p Fn 1906 4259 a(R) p Fk 2009 4259 a(\025) p Fo 2114 4259 a(1.) p 2234 4259 a(Th) m(us) p 2481 4259 a(w) m(e) p 2624 4259 a(ha) m(v) m(e) p Fk 597 4454 a(jj) p Fn(A) p Fm 726 4469 a(N) p Fo 793 4454 a(\() p Fn(w) p Fo 904 4454 a(\)) p Fn(\036) p Fk(jj) p Fl 1056 4413 a(2) p Fm 1056 4479 a(N) p Fk 1205 4454 a(\024) p Fi 1416 4371 a(X) p Fm 1365 4555 a(m) p Fe 1427 4565 a(j) p Fg 1460 4555 a(2) p Fl(\000) p Fd 1551 4564 a(1) p Fk 1603 4454 a(jj) p Fn(t) p Fm 1694 4469 a(z) p Fe 1727 4479 a(j) p Fn 1763 4454 a(\021) p Fl 1815 4400 a(\() p Fm(m) p Fe 1904 4410 a(j) p Fl 1937 4400 a(\)) 1811 4476 y(0) p Fo 1968 4454 a(\() p Fn(H) p Fl 2087 4469 a(1) p Fk 2149 4454 a(\000) p Fn 2248 4454 a(w) p Fo 2321 4454 a(\)) p Fg 2359 4413 a(\000) p Fl(1) p Fn 2453 4454 a(t) p Fg 2488 4469 a(\000) p Fm(z) p Fe 2576 4479 a(j) p Fn 2613 4454 a(\021) p Fm 2661 4469 a(m) p Fe 2723 4479 a(j) p Fn 2760 4454 a(\036) p Fk(jj) p Fl 2874 4413 a(2) p Fm 2874 4479 a(N) p Fo 1365 4717 a(+) p Fi 1522 4634 a(X) p Fm 1458 4819 a(\014) s(;\015) p Fl 1561 4819 a(=0) p Fm(;) p Fl(1) p Fi 1812 4634 a(X) p Fm 1722 4818 a(m) p Fg(2) p Fl(\000) p Fd 1875 4830 a(0) p Fe(;\014) s(;\015) p Fk 2038 4717 a(jj) p Fn(t) p Fm 2129 4732 a(m) p Fn 2195 4717 a(\021) p Fl 2247 4666 a(\() p Fm(m) p Fl(\)) 2243 4739 y(0) p Fo 2368 4717 a(\() p Fn(H) p Fl 2487 4732 a(0) p Fk 2549 4717 a(\000) p Fn 2648 4717 a(w) p Fo 2721 4717 a(\)) p Fg 2759 4676 a(\000) p Fl(1) p Fn 2853 4717 a(t) p Fg 2888 4732 a(\000) p Fm(m) p Fn 3010 4717 a(\021) p Fm 3058 4732 a(m) p Fn 3124 4717 a(\036) p Fk(jj) p Fl 3238 4676 a(2) p Fm 3238 4742 a(N) p Fk 1205 4965 a(\024) p Fn 1365 4965 a(C) p Fl 1435 4980 a(10) p Fk 1510 4965 a(jj) p Fn(\036) p Fk(jj) p Fl 1680 4924 a(2) p Fn 1718 4965 a(;) p Fo 1794 4965 a(if) p Fn 1890 4965 a(R) p Fk 1993 4965 a(\025) p Fo 2098 4965 a(1) p Fn(;) p Fo 1902 5214 a(56) p eop %%Page: 57 57 57 56 bop Fo 0 631 a(where) p Fn 370 888 a(C) p Fl 440 903 a(10) p Fo 542 888 a(=) p 646 888 a(4) p Fn(C) p Fl 765 903 a(1) p Fo 804 888 a(\(1) p 912 888 a(+) p Fn 1010 888 a(E) p Fo 1111 888 a(+) p Fn 1209 888 a(\016) p Fl 1252 903 a(0) p Fo 1291 888 a(\)) p Fl 1329 847 a(2) p Fi 1385 742 a( ) p Fo 1451 888 a(1) p 1522 888 a(+) p 1708 888 a(sup) p Fg 1620 971 a(j) p Fm(w) p Fg 1693 971 a(\000) p Fm(E) p Fg 1804 971 a(j) p Fl(=) p Fm(\016) p Fd 1910 980 a(0) p Fo 1962 888 a(max) p Fm 1959 945 a(n) p Fl(=0) p Fm(;) p Fl(1) p Fi 2164 792 a(\020) p Fk 2213 888 a(jj) p Fo(\() p Fn(H) p Fm 2388 903 a(n) p Fk 2456 888 a(\000) p Fn 2556 888 a(w) p Fo 2629 888 a(\)) p Fg 2667 847 a(\000) p Fl(1) p Fk 2761 888 a(jj) p Fl 2817 847 a(2) p Fm 2817 913 a(B) p Fl 2873 913 a(\() p Fm(L) p Fd 2948 894 a(2) p Fl 2983 913 a(\() p Fa(R) p Fd 3071 894 a(2) p Fl 3105 913 a(\);) p Fm(L) p Fd 3200 894 a(2) p Fl 3235 913 a(\() p Fa(R) p Fd 3323 894 a(2) p Fl 3358 913 a(\)\)) p Fi 3417 792 a(\021) 3466 742 y(!) p Fo 0 1189 a(and) p 190 1189 a(w) m(e) p 333 1189 a(used) p 556 1189 a(the) p 724 1189 a(fact) p 916 1189 a(#) p Fk(f) p Fn(m) p Fk 1161 1189 a(2) p Fo 1255 1189 a(\000;) p Fn 1360 1189 a(z) p Fk 1437 1189 a(2) p Fo 1531 1189 a(supp) p Fn 1748 1189 a(\021) p Fm 1796 1204 a(m) p Fk 1863 1189 a(g) p 1941 1189 a(\024) p Fo 2046 1189 a(4) p 2127 1189 a(for) p 2276 1189 a(an) m(y) p Fn 2460 1189 a(z) p Fk 2538 1189 a(2) p Fj 2632 1189 a(R) p Fl 2719 1147 a(2) p Fo 2759 1189 a(.) p 2829 1121 64 4 v 2829 1214 4 93 v 2889 1214 V 2829 1217 64 4 v 146 1417 a(W) p 238 1417 a(e) p 314 1417 a(ha) m(v) m(e) 535 1637 y(\() p Fn(H) p Fm 654 1652 a(N) p Fk 743 1637 a(\000) p Fn 842 1637 a(w) p Fo 915 1637 a(\)) p Fn(A) p Fm 1026 1652 a(N) p Fo 1093 1637 a(\() p Fn(w) p Fo 1204 1637 a(\)) p 1324 1637 a(=) p Fi 1534 1554 a(X) p Fm 1483 1738 a(m) p Fe 1545 1748 a(j) p Fg 1578 1738 a(2) p Fl(\000) p Fd 1669 1747 a(1) p Fo 1704 1637 a(\() p Fn(H) p Fm 1823 1652 a(N) p Fk 1912 1637 a(\000) p Fn 2012 1637 a(w) p Fo 2085 1637 a(\)) p Fn(t) p Fm 2158 1652 a(z) p Fe 2191 1662 a(j) p Fn 2227 1637 a(\021) p Fl 2279 1583 a(\() p Fm(m) p Fe 2368 1593 a(j) p Fl 2401 1583 a(\)) 2275 1659 y(0) p Fo 2433 1637 a(\() p Fn(H) p Fl 2552 1652 a(1) p Fk 2613 1637 a(\000) p Fn 2713 1637 a(w) p Fo 2786 1637 a(\)) p Fg 2824 1596 a(\000) p Fl(1) p Fn 2917 1637 a(t) p Fg 2952 1652 a(\000) p Fm(z) p Fe 3040 1662 a(j) p Fn 3077 1637 a(\021) p Fm 3125 1652 a(m) p Fe 3187 1662 a(j) p Fo 1483 1900 a(+) p Fi 1610 1817 a(X) p Fm 1576 2001 a(m) p Fg(2) p Fl(\000) p Fd 1729 2010 a(0) p Fo 1764 1900 a(\() p Fn(H) p Fm 1883 1915 a(N) p Fk 1972 1900 a(\000) p Fn 2072 1900 a(w) p Fo 2145 1900 a(\)) p Fn(t) p Fm 2218 1915 a(m) p Fn 2284 1900 a(\021) p Fl 2336 1849 a(\() p Fm(m) p Fl(\)) 2332 1922 y(0) p Fo 2457 1900 a(\() p Fn(H) p Fl 2576 1915 a(0) p Fk 2637 1900 a(\000) p Fn 2737 1900 a(w) p Fo 2810 1900 a(\)) p Fg 2848 1859 a(\000) p Fl(1) p Fn 2942 1900 a(t) p Fg 2977 1915 a(\000) p Fm(m) p Fn 3098 1900 a(\021) p Fm 3146 1915 a(m) p Fo 1324 2156 a(=) p Fi 1534 2073 a(X) p Fm 1483 2256 a(m) p Fe 1545 2266 a(j) p Fg 1578 2256 a(2) p Fl(\000) p Fd 1669 2265 a(1) p Fo 1704 2156 a(\() p Fn(t) p Fm 1777 2171 a(z) p Fe 1810 2181 a(j) p Fo 1847 2156 a([) p Fn(H) p Fl 1955 2171 a(1) p Fn 1994 2156 a(;) p 2038 2156 a(\021) p Fl 2090 2101 a(\() p Fm(m) p Fe 2179 2111 a(j) p Fl 2212 2101 a(\)) 2086 2177 y(0) p Fo 2243 2156 a(]\() p Fn(H) p Fl 2389 2171 a(1) p Fk 2451 2156 a(\000) p Fn 2550 2156 a(w) p Fo 2623 2156 a(\)) p Fg 2661 2114 a(\000) p Fl(1) p Fn 2755 2156 a(t) p Fg 2790 2171 a(\000) p Fm(z) p Fe 2878 2181 a(j) p Fn 2915 2156 a(\021) p Fm 2963 2171 a(m) p Fe 3025 2181 a(j) p Fo 3084 2156 a(+) p Fn 3182 2156 a(\021) p Fl 3234 2114 a(2) p Fm 3230 2180 a(m) p Fe 3292 2190 a(j) p Fo 3329 2156 a(\)) 1483 2418 y(+) p Fi 1610 2335 a(X) p Fm 1576 2519 a(m) p Fg(2) p Fl(\000) p Fd 1729 2528 a(0) p Fo 1764 2418 a(\() p Fn(t) p Fm 1837 2433 a(m) p Fo 1904 2418 a([) p Fn(H) p Fl 2012 2433 a(0) p Fn 2051 2418 a(;) p 2095 2418 a(\021) p Fl 2147 2367 a(\() p Fm(m) p Fl(\)) 2143 2440 y(0) p Fo 2268 2418 a(]\() p Fn(H) p Fl 2414 2433 a(0) p Fk 2475 2418 a(\000) p Fn 2575 2418 a(w) p Fo 2648 2418 a(\)) p Fg 2686 2377 a(\000) p Fl(1) p Fn 2780 2418 a(t) p Fg 2815 2433 a(\000) p Fm(m) p Fn 2936 2418 a(\021) p Fm 2984 2433 a(m) p Fo 3073 2418 a(+) p Fn 3171 2418 a(\021) p Fl 3223 2377 a(2) p Fm 3219 2443 a(m) p Fo 3286 2418 a(\)) 1324 2645 y(=) p 1483 2645 a(1) p 1554 2645 a(+) p Fn 1652 2645 a(T) p Fm 1709 2660 a(N) p Fo 1777 2645 a(\() p Fn(w) p Fo 1888 2645 a(\)) p Fn(;) p Fo 3653 2645 a(\(8.30\)) 0 2865 y(where) p Fn 907 3085 a(T) p Fm 964 3100 a(N) p Fo 1031 3085 a(\() p Fn(w) p Fo 1142 3085 a(\)) p 1263 3085 a(=) p Fi 1472 3002 a(X) p Fm 1422 3185 a(m) p Fe 1484 3195 a(j) p Fg 1517 3185 a(2) p Fl(\000) p Fd 1608 3194 a(1) p Fn 1659 3085 a(t) p Fm 1694 3100 a(z) p Fe 1727 3110 a(j) p Fo 1764 3085 a([) p Fn(H) p Fl 1872 3100 a(1) p Fn 1911 3085 a(;) p 1955 3085 a(\021) p Fl 2007 3030 a(\() p Fm(m) p Fe 2096 3040 a(j) p Fl 2129 3030 a(\)) 2003 3106 y(0) p Fo 2160 3085 a(]\() p Fn(H) p Fl 2306 3100 a(1) p Fk 2368 3085 a(\000) p Fn 2467 3085 a(w) p Fo 2540 3085 a(\)) p Fg 2578 3043 a(\000) p Fl(1) p Fn 2672 3085 a(t) p Fg 2707 3100 a(\000) p Fm(z) p Fe 2795 3110 a(j) p Fn 2832 3085 a(\021) p Fm 2880 3100 a(m) p Fe 2942 3110 a(j) p Fo 1422 3347 a(+) p Fi 1548 3264 a(X) p Fm 1514 3448 a(m) p Fg(2) p Fl(\000) p Fd 1667 3457 a(0) p Fn 1719 3347 a(t) p Fm 1754 3362 a(m) p Fo 1821 3347 a([) p Fn(H) p Fl 1929 3362 a(0) p Fn 1968 3347 a(;) p 2012 3347 a(\021) p Fl 2064 3296 a(\() p Fm(m) p Fl(\)) 2060 3369 y(0) p Fo 2185 3347 a(]\() p Fn(H) p Fl 2331 3362 a(0) p Fk 2392 3347 a(\000) p Fn 2492 3347 a(w) p Fo 2565 3347 a(\)) p Fg 2603 3306 a(\000) p Fl(1) p Fn 2697 3347 a(t) p Fg 2732 3362 a(\000) p Fm(m) p Fn 2853 3347 a(\021) p Fm 2901 3362 a(m) p Fn 2968 3347 a(:) p Fo 3653 3347 a(\(8.31\)) 0 3653 y(W) p 92 3653 a(e) p 173 3653 a(used) p 400 3653 a(the) p 573 3653 a(in) m(tert) m(wining) p 1125 3653 a(prop) s(ert) m(y) p 1528 3653 a(of) p Fn 1644 3653 a(t) p Fg 1679 3668 a(\000) p Fm(m) p Fo 1838 3653 a(and) p 2033 3653 a(the) p 2206 3653 a(equalit) m(y) p Fi 2582 3586 a(P) p Fm 2669 3673 a(m) p Fg(2) p Fl(\000) p Fn 2844 3653 a(\021) p Fl 2896 3616 a(2) p Fm 2892 3677 a(m) p Fo 2994 3653 a(=) p 3106 3653 a(1.) p 3239 3653 a(Notice) p 3550 3653 a(that) p 3767 3653 a(the) 0 3790 y(op) s(erator) p 385 3790 a([) p Fn(H) p Fl 493 3805 a(1) p Fn 532 3790 a(;) p 576 3790 a(\021) p Fl 628 3736 a(\() p Fm(m) p Fe 717 3746 a(j) p Fl 750 3736 a(\)) 624 3811 y(0) p Fo 781 3790 a(]\() p Fn(H) p Fl 927 3805 a(1) p Fk 972 3790 a(\000) p Fn 1054 3790 a(w) p Fo 1127 3790 a(\)) p Fg 1165 3754 a(\000) p Fl(1) p Fo 1284 3790 a(is) p 1374 3790 a(w) m(ell-de\014ned) p 1899 3790 a(b) m(y) p 2026 3790 a(Lemma) p 2366 3790 a(8.2,) p 2544 3790 a(and) p 2725 3790 a(the) p 2885 3790 a(op) s(erator) p 3270 3790 a([) p Fn(H) p Fl 3378 3805 a(0) p Fn 3417 3790 a(;) p 3461 3790 a(\021) p Fl 3513 3739 a(\() p Fm(m) p Fl(\)) 3509 3811 y(0) p Fo 3634 3790 a(]\() p Fn(H) p Fl 3780 3805 a(0) p Fk 3825 3790 a(\000) p Fn 0 3910 a(w) p Fo 73 3910 a(\)) p Fg 111 3874 a(\000) p Fl(1) p Fo 237 3910 a(is) p 335 3910 a(w) m(ell-de\014ned) p 869 3910 a(b) m(y) p 1005 3910 a(Lemma) p 1353 3910 a(8.3.) p Fq 146 4113 a(Lemma) p 543 4113 a(8.10) p Ff 791 4113 a(Two) p 1008 4113 a(sums) p 1252 4113 a(in) p 1369 4113 a(the) p 1528 4113 a(right) p 1758 4113 a(hand) p 1995 4113 a(side) p 2192 4113 a(of) p 2304 4113 a(\(8.31\)) p 2595 4113 a(c) p 2635 4113 a(onver) p 2866 4113 a(ge) p 2987 4113 a(in) p 3104 4113 a(the) p 3263 4113 a(str) p 3371 4113 a(ong) p 3553 4113 a(op) p 3648 4113 a(er) p 3729 4113 a(ator) 0 4234 y(top) p 127 4234 a(olo) p 247 4234 a(gy) p 385 4234 a(of) p 512 4234 a(the) p 685 4234 a(b) p 725 4234 a(ounde) p 972 4234 a(d) p 1068 4234 a(op) p 1163 4234 a(er) p 1244 4234 a(ator) p 1463 4234 a(fr) p 1529 4234 a(om) p Fn 1706 4234 a(L) p Fl 1772 4198 a(2) p Fo 1811 4234 a(\() p Fj(R) p Fl 1937 4192 a(2) p Fo 1976 4234 a(\)) p Ff 2061 4234 a(to) p Fn 2189 4234 a(L) p Fl 2255 4198 a(2) p Fo 2295 4234 a(\() p Fj(R) p Fl 2421 4192 a(2) p Fo 2460 4234 a(\)) p Ff(.) p 2608 4234 a(Mor) p 2781 4234 a(e) p 2821 4234 a(over,) p 3081 4234 a(ther) p 3244 4234 a(e) p 3336 4234 a(is) p 3452 4234 a(a) p 3548 4234 a(c) p 3588 4234 a(onstant) p Fn 0 4354 a(C) p Fl 70 4369 a(11) p Fn 172 4354 a(>) p Fo 276 4354 a(0) p Ff 359 4354 a(dep) p 499 4354 a(endent) p 815 4354 a(only) p 1027 4354 a(on) p Fn 1166 4354 a(B) p Ff 1245 4354 a(,) p Fn 1310 4354 a(E) p Ff 1388 4354 a(,) p Fn 1453 4354 a(\016) p Fl 1496 4369 a(0) p Ff 1571 4354 a(and) p Fn 1760 4354 a(\013) p Ff 1857 4354 a(such) p 2079 4354 a(that) p Fk 1310 4617 a(jj) p Fn(T) p Fm 1423 4632 a(N) p Fo 1490 4617 a(\() p Fn(w) p Fo 1601 4617 a(\)) p Fk(jj) p Fm 1695 4634 a(B) p Fl 1751 4634 a(\() p Fm(L) p Fd 1826 4615 a(2) p Fl 1860 4634 a(\() p Fa(R) p Fd 1948 4615 a(2) p Fl 1983 4634 a(\);) p Fm(L) p Fd 2078 4615 a(2) p Fl 2113 4634 a(\() p Fa(R) p Fd 2201 4615 a(2) p Fl 2235 4634 a(\)\)) p Fk 2322 4617 a(\024) p Fn 2437 4550 a(C) p Fl 2507 4565 a(11) p 2437 4594 145 4 v Fn 2472 4685 a(R) p Ff 0 4868 a(for) p 156 4868 a(every) p Fn 413 4868 a(w) p Fk 514 4868 a(2) p Fj 608 4868 a(C) p Ff 731 4868 a(with) p Fk 943 4868 a(j) p Fn(w) p Fk 1065 4868 a(\000) p Fn 1164 4868 a(E) p Fk 1242 4868 a(j) p Fo 1298 4868 a(=) p Fn 1401 4868 a(\016) p Fl 1444 4883 a(0) p Ff 1484 4868 a(,) p 1549 4868 a(if) p Fn 1643 4868 a(R) p Fk 1746 4868 a(\025) p Fo 1851 4868 a(1) p Ff(.) p Fo 1902 5214 a(57) p eop %%Page: 58 58 58 57 bop Ff 146 631 a(Pr) p 248 631 a(o) p 293 631 a(of.) p Fo 436 631 a(By) p 589 631 a(the) p 757 631 a(similar) p 1078 631 a(argumen) m(t) p 1514 631 a(in) p 1628 631 a(the) p 1797 631 a(pro) s(of) p 2052 631 a(of) p 2163 631 a(Lemma) p 2511 631 a(8.9,) p 2696 631 a(w) m(e) p 2840 631 a(see) p 2998 631 a(that) p 3209 631 a(it) p 3307 631 a(is) p 3405 631 a(su\016cien) m(t) p 3815 631 a(to) 0 751 y(sho) m(w) p 242 751 a(that) p 453 751 a(there) p 702 751 a(is) p 800 751 a(a) p 882 751 a(constan) m(t) p Fn 1275 751 a(C) p Fg 1352 715 a(0) p Fn 1403 751 a(>) p Fo 1507 751 a(0) p 1588 751 a(dep) s(enden) m(t) p 2059 751 a(only) p 2274 751 a(on) p Fn 2409 751 a(B) p Fo 2488 751 a(,) p Fn 2548 751 a(E) p Fo 2626 751 a(,) p Fn 2685 751 a(\016) p Fl 2728 766 a(0) p Fo 2801 751 a(and) p Fn 2990 751 a(\013) p Fo 3085 751 a(suc) m(h) p 3305 751 a(that) p Fk 673 985 a(jj) p Fn(t) p Fm 764 1000 a(z) p Fe 797 1010 a(j) p Fo 833 985 a([) p Fn(H) p Fl 941 1000 a(1) p Fn 981 985 a(;) p 1025 985 a(\021) p Fl 1077 931 a(\() p Fm(m) p Fe 1166 941 a(j) p Fl 1199 931 a(\)) 1073 1007 y(0) p Fo 1230 985 a(]\() p Fn(H) p Fl 1376 1000 a(1) p Fk 1438 985 a(\000) p Fn 1537 985 a(w) p Fo 1610 985 a(\)) p Fg 1648 944 a(\000) p Fl(1) p Fn 1742 985 a(t) p Fg 1777 1000 a(\000) p Fm(z) p Fe 1865 1010 a(j) p Fn 1902 985 a(\021) p Fm 1950 1000 a(m) p Fe 2012 1010 a(j) p Fn 2049 985 a(\036) p Fk(jj) p Fl 2163 944 a(2) p Fk 2284 985 a(\024) p Fn 2462 917 a(C) p Fg 2539 881 a(0) p 2455 962 115 4 v Fn 2455 1053 a(R) p Fl 2530 1024 a(2) p Fk 2579 985 a(jj) p Fn(\036) p Fk(jj) p Fl 2749 944 a(2) p Fn 2885 985 a(m) p Fm 2970 1000 a(j) p Fk 3034 985 a(2) p Fo 3128 985 a(\000) p Fl 3189 1000 a(1) p Fo 3653 985 a(\(8.32\)) p Fk 716 1227 a(jj) p Fn(t) p Fm 807 1242 a(m) p Fo 874 1227 a([) p Fn(H) p Fl 982 1242 a(0) p Fn 1021 1227 a(;) p 1065 1227 a(\021) p Fl 1117 1176 a(\() p Fm(m) p Fl(\)) 1113 1249 y(0) p Fo 1238 1227 a(]\() p Fn(H) p Fm 1384 1242 a(N) p Fk 1473 1227 a(\000) p Fn 1573 1227 a(w) p Fo 1646 1227 a(\)) p Fg 1684 1186 a(\000) p Fl(1) p Fn 1777 1227 a(t) p Fg 1812 1242 a(\000) p Fm(m) p Fn 1934 1227 a(\021) p Fm 1982 1242 a(m) p Fn 2049 1227 a(\036) p Fk(jj) p Fl 2163 1186 a(2) p Fk 2284 1227 a(\024) p Fn 2462 1159 a(C) p Fg 2539 1123 a(0) p 2455 1204 V Fn 2455 1295 a(R) p Fl 2530 1266 a(2) p Fk 2579 1227 a(jj) p Fn(\036) p Fk(jj) p Fl 2749 1186 a(2) p Fn 2885 1227 a(m) p Fk 2998 1227 a(2) p Fo 3092 1227 a(\000) p Fl 3153 1242 a(0) p Fo 3653 1227 a(\(8.33\)) 0 1454 y(for) p 152 1454 a(ev) m(ery) p Fn 411 1454 a(\036) p Fk 501 1454 a(2) p Fn 599 1454 a(C) p Fg 676 1418 a(1) p Fl 669 1478 a(0) p Fo 751 1454 a(\() p Fj(R) p Fl 876 1412 a(2) p Fo 916 1454 a(\)) p 989 1454 a(and) p 1181 1454 a(ev) m(ery) p Fn 1441 1454 a(w) p Fk 1545 1454 a(2) p Fj 1643 1454 a(C) p Fo 1767 1454 a(with) p Fk 1991 1454 a(j) p Fn(w) p Fk 2115 1454 a(\000) p Fn 2217 1454 a(E) p Fk 2295 1454 a(j) p Fo 2354 1454 a(=) p Fn 2462 1454 a(\016) p Fl 2505 1469 a(0) p Fo 2545 1454 a(,) p 2607 1454 a(if) p Fn 2699 1454 a(R) p Fk 2806 1454 a(\025) p Fo 2916 1454 a(1.) p 3042 1454 a(By) p 3198 1454 a(\(4.5\)) p 3433 1454 a(and) p 3626 1454 a(\(8.28\),) 0 1574 y(w) m(e) p 144 1574 a(ha) m(v) m(e) p Fk 640 1760 a(jj) p Fn(t) p Fm 731 1775 a(z) p Fe 764 1785 a(j) p Fo 800 1760 a([) p Fn(H) p Fl 908 1775 a(1) p Fn 948 1760 a(;) p 992 1760 a(\021) p Fl 1044 1706 a(\() p Fm(m) p Fe 1133 1716 a(j) p Fl 1166 1706 a(\)) 1040 1782 y(0) p Fo 1197 1760 a(]\() p Fn(H) p Fl 1343 1775 a(1) p Fk 1405 1760 a(\000) p Fn 1504 1760 a(w) p Fo 1577 1760 a(\)) p Fg 1615 1719 a(\000) p Fl(1) p Fn 1709 1760 a(t) p Fg 1744 1775 a(\000) p Fm(z) p Fe 1832 1785 a(j) p Fn 1869 1760 a(\021) p Fm 1917 1775 a(m) p Fe 1979 1785 a(j) p Fn 2016 1760 a(\036) p Fk(jj) p Fl 2130 1719 a(2) p Fo 481 1932 a(=) p Fk 640 1932 a(jj) p Fo([) p Fn(H) p Fl 804 1947 a(1) p Fn 843 1932 a(;) p 887 1932 a(\021) p Fl 939 1878 a(\() p Fm(m) p Fe 1028 1888 a(j) p Fl 1061 1878 a(\)) 935 1954 y(0) p Fo 1093 1932 a(]\() p Fn(H) p Fl 1239 1947 a(1) p Fk 1300 1932 a(\000) p Fn 1400 1932 a(w) p Fo 1473 1932 a(\)) p Fg 1511 1891 a(\000) p Fl(1) p Fn 1604 1932 a(t) p Fg 1639 1947 a(\000) p Fm(z) p Fe 1727 1957 a(j) p Fn 1764 1932 a(\021) p Fm 1812 1947 a(m) p Fe 1874 1957 a(j) p Fn 1911 1932 a(\036) p Fk(jj) p Fl 2025 1891 a(2) p Fk 480 2077 a(\024) p Fn 640 2077 a(C) p Fl 710 2092 a(0) p Fk 750 2077 a(jj) p Fo(\() p Fn(H) p Fl 925 2092 a(1) p Fk 985 2077 a(\000) p Fn 1085 2077 a(w) p Fo 1158 2077 a(\)) p Fg 1196 2036 a(\000) p Fl(1) p Fn 1290 2077 a(t) p Fg 1325 2092 a(\000) p Fm(z) p Fe 1413 2102 a(j) p Fn 1449 2077 a(\021) p Fm 1497 2092 a(m) p Fe 1559 2102 a(j) p Fn 1596 2077 a(\036) p Fk(jj) p Fl 1710 2036 a(2) 1710 2102 y(1) p Fk 480 2280 a(\024) p Fo 650 2213 a(10) p Fn(C) p 650 2257 175 4 v 680 2349 a(R) p Fl 755 2320 a(2) p Fo 835 2280 a(\(1) p 944 2280 a(+) p Fn 1042 2280 a(E) p Fo 1142 2280 a(+) p Fn 1240 2280 a(\016) p Fl 1283 2295 a(0) p Fo 1323 2280 a(\)) p Fl 1361 2239 a(2) p Fo 1400 2280 a(\(1) p 1509 2280 a(+) p Fk 1607 2280 a(jj) p Fo(\() p Fn(H) p Fl 1782 2295 a(1) p Fk 1842 2280 a(\000) p Fn 1942 2280 a(w) p Fo 2015 2280 a(\)) p Fg 2053 2239 a(\000) p Fl(1) p Fk 2147 2280 a(jj) p Fo(\)) p Fm 2241 2297 a(B) p Fl 2297 2297 a(\() p Fm(L) p Fd 2372 2278 a(2) p Fl 2407 2297 a(\() p Fa(R) p Fd 2495 2278 a(2) p Fl 2529 2297 a(\);) p Fm(L) p Fd 2624 2278 a(2) p Fl 2659 2297 a(\() p Fa(R) p Fd 2747 2278 a(2) p Fl 2782 2297 a(\)\)) p Fk 2841 2280 a(jj) p Fn(\036) p Fk(jj) p Fl 3011 2239 a(2) p Fn 3049 2280 a(;) p Fo 3125 2280 a(if) p Fn 3221 2280 a(R) p Fk 3324 2280 a(\025) p Fo 3429 2280 a(1) p Fn(;) p Fo 0 2501 a(where) p Fn 283 2501 a(C) p Fl 353 2516 a(0) p Fo 422 2501 a(=) p 527 2501 a(10) p 642 2501 a(sup\() p Fk(jr) p Fn(\021) p Fm 985 2516 a(m) p Fe 1047 2526 a(j) p Fk 1084 2501 a(j) p Fl 1112 2465 a(2) p Fo 1174 2501 a(+) p Fk 1272 2501 a(jr) p Fo(\() p Fn(@) p Fm 1472 2516 a(x) p Fn 1516 2501 a(\021) p Fm 1564 2516 a(m) p Fe 1626 2526 a(j) p Fo 1664 2501 a(\)) p Fk(j) p Fl 1730 2465 a(2) p Fo 1791 2501 a(+) p Fk 1890 2501 a(jr) p Fo(\() p Fn(@) p Fm 2090 2516 a(y) p Fn 2132 2501 a(\021) p Fm 2180 2516 a(m) p Fe 2242 2526 a(j) p Fo 2279 2501 a(\)) p Fk(j) p Fl 2345 2465 a(2) p Fo 2384 2501 a(\).) p 2495 2501 a(Th) m(us) p 2744 2501 a(\(8.32\)) p 3026 2501 a(holds.) p 3323 2501 a(W) p 3415 2501 a(e) p 3492 2501 a(can) p 3672 2501 a(pro) m(v) m(e) 0 2622 y(\(8.33\)) p 282 2622 a(similarly) p 644 2622 a(.) p 710 2554 64 4 v 710 2647 4 93 v 771 2647 V 710 2650 64 4 v 146 2811 a(T) p 208 2811 a(ak) m(e) p 380 2811 a(a) p 459 2811 a(large) p 695 2811 a(n) m(um) m(b) s(er) p Fn 1047 2811 a(R) p Fg 1122 2775 a(0) p Fn 1173 2811 a(>) p 1277 2811 a(R) p Fl 1351 2826 a(0) p Fo 1421 2811 a(suc) m(h) p 1639 2811 a(that) p Fm 1858 2772 a(C) p Fd 1908 2781 a(11) p 1858 2788 116 4 v Fm 1877 2845 a(R) p Fb 1930 2826 a(0) p Fn 2011 2811 a(<) p Fl 2124 2772 a(1) p 2124 2788 36 4 v 2124 2845 a(2) p Fo 2169 2811 a(.) p 2239 2811 a(By) p 2390 2811 a(Lemma) p 2736 2811 a(8.10,) p 2967 2811 a(w) m(e) p 3108 2811 a(ha) m(v) m(e) p Fk 3331 2811 a(jj) p Fn(T) p Fm 3444 2826 a(N) p Fo 3510 2811 a(\() p Fn(w) p Fo 3621 2811 a(\)) p Fk(jj) p 3742 2811 a(\024) p Fl 3857 2772 a(1) p 3857 2788 V 3857 2845 a(2) p Fo 0 2931 a(and) p 192 2931 a(th) m(us) p 408 2931 a(1) p 480 2931 a(+) p Fn 580 2931 a(T) p Fm 637 2946 a(N) p Fo 704 2931 a(\() p Fn(w) p Fo 815 2931 a(\)) p 887 2931 a(is) p 987 2931 a(in) m(v) m(ertible) p 1420 2931 a(for) p Fn 1571 2931 a(w) p Fk 1674 2931 a(2) p Fn 1772 2931 a(C) p Fo 1883 2931 a(with) p Fk 2107 2931 a(j) p Fn(w) p Fk 2231 2931 a(\000) p Fn 2332 2931 a(E) p Fk 2410 2931 a(j) p Fo 2468 2931 a(=) p Fn 2575 2931 a(\016) p Fl 2618 2946 a(0) p Fo 2658 2931 a(,) p 2720 2931 a(if) p Fn 2811 2931 a(R) p Fk 2917 2931 a(\025) p Fn 3026 2931 a(R) p Fg 3101 2895 a(0) p Fo 3124 2931 a(.) p 3200 2931 a(Then) p Fn 3457 2931 a(H) p Fm 3538 2946 a(N) p Fk 3629 2931 a(\000) p Fn 3730 2931 a(w) p Fo 3836 2931 a(is) 0 3052 y(also) p 196 3052 a(in) m(v) m(ertible) p 626 3052 a(and) p 816 3052 a(its) p 952 3052 a(in) m(v) m(erse) p 1275 3052 a(is) p 1373 3052 a(giv) m(en) p 1628 3052 a(b) m(y) 824 3237 y(\() p Fn(H) p Fm 943 3252 a(N) p Fk 1032 3237 a(\000) p Fn 1132 3237 a(w) p Fo 1205 3237 a(\)) p Fg 1243 3196 a(\000) p Fl(1) p Fo 1364 3237 a(=) p Fn 1467 3237 a(A) p Fm 1540 3252 a(N) p Fo 1608 3237 a(\() p Fn(w) p Fo 1719 3237 a(\)) p Fk 1778 3237 a(\000) p Fn 1878 3237 a(A) p Fm 1951 3252 a(N) p Fo 2018 3237 a(\() p Fn(w) p Fo 2129 3237 a(\)) p Fn(T) p Fm 2224 3252 a(N) p Fo 2291 3237 a(\() p Fn(w) p Fo 2402 3237 a(\)\(1) p 2548 3237 a(+) p Fn 2646 3237 a(T) p Fm 2703 3252 a(N) p Fo 2771 3237 a(\() p Fn(w) p Fo 2882 3237 a(\)\)) p Fg 2958 3196 a(\000) p Fl(1) p Fn 3051 3237 a(:) p Fo 3653 3237 a(\(8.34\)) 0 3423 y(In) m(tegrating) p 504 3423 a(\(8.34\)) p 786 3423 a(on) p Fk 921 3423 a(f) p Fn(w) p Fo 1044 3423 a(;) p Fk 1088 3423 a(j) p Fn(w) p Fk 1209 3423 a(\000) p Fn 1309 3423 a(E) p Fk 1387 3423 a(j) p Fo 1442 3423 a(=) p Fn 1546 3423 a(\016) p Fl 1589 3438 a(0) p Fk 1629 3423 a(g) p Fo(,) p 1738 3423 a(w) m(e) p 1882 3423 a(ha) m(v) m(e) p Fn 610 3626 a(P) p Fl 673 3641 a(\() p Fm(E) p Fg 756 3641 a(\000) p Fm(\016) p Fd 842 3650 a(0) p Fm 876 3641 a(;E) p Fl 952 3641 a(+) p Fm(\016) p Fd 1038 3650 a(0) p Fl 1072 3641 a(\)) p Fo 1104 3626 a(\() p Fn(H) p Fm 1223 3641 a(N) p Fo 1290 3626 a(\)) p 1355 3626 a(=) p Fi 1509 3543 a(X) p Fm 1459 3726 a(m) p Fe 1521 3736 a(j) p Fg 1554 3726 a(2) p Fl(\000) p Fd 1645 3735 a(1) p Fn 1696 3626 a(t) p Fm 1731 3641 a(z) p Fe 1764 3651 a(j) p Fn 1801 3626 a(\021) p Fl 1853 3572 a(\() p Fm(m) p Fe 1942 3582 a(j) p Fl 1975 3572 a(\)) 1849 3647 y(0) p Fn 2007 3626 a(P) p Fl 2070 3641 a(\() p Fm(E) p Fg 2153 3641 a(\000) p Fm(\016) p Fd 2239 3650 a(0) p Fm 2273 3641 a(;E) p Fl 2349 3641 a(+) p Fm(\016) p Fd 2435 3650 a(0) p Fl 2469 3641 a(\)) p Fo 2500 3626 a(\() p Fn(H) p Fl 2619 3641 a(1) p Fo 2659 3626 a(\)) p Fn(t) p Fg 2732 3641 a(\000) p Fm(z) p Fe 2820 3651 a(j) p Fn 2856 3626 a(\021) p Fm 2904 3641 a(m) p Fe 2966 3651 a(j) p Fo 3025 3626 a(+) p Fn 3123 3626 a(R) p Fm 3197 3641 a(N) p Fn 3265 3626 a(;) p Fo 3653 3626 a(\(8.35\)) 0 3897 y(where) p Fn 973 4018 a(R) p Fm 1047 4033 a(N) p Fo 1142 4018 a(=) p Fk 1245 4018 a(\000) p Fi 1339 3901 a(I) p Fg 1386 4089 a(j) p Fm(w) p Fg 1459 4089 a(\000) p Fm(E) p Fg 1570 4089 a(j) p Fl(=) p Fm(\016) p Fd 1676 4098 a(0) p Fn 1729 4018 a(A) p Fm 1802 4033 a(N) p Fo 1869 4018 a(\() p Fn(w) p Fo 1980 4018 a(\)) p Fn(T) p Fm 2075 4033 a(N) p Fo 2142 4018 a(\() p Fn(w) p Fo 2253 4018 a(\)\(1) p 2399 4018 a(+) p Fn 2497 4018 a(T) p Fm 2554 4033 a(N) p Fo 2622 4018 a(\() p Fn(w) p Fo 2733 4018 a(\)\)) p Fg 2809 3977 a(\000) p Fl(1) p Fn 2902 4018 a(:) p Fo 0 4227 a(By) p 151 4227 a(Lemma) p 496 4227 a(8.9) p 651 4227 a(and) p 838 4227 a(Lemma) p 1184 4227 a(8.10,) p 1415 4227 a(there) p 1662 4227 a(exist) p 1890 4227 a(a) p 1969 4227 a(constan) m(t) p Fn 2360 4227 a(R) p Fl 2434 4242 a(2) p Fn 2501 4227 a(>) p 2605 4227 a(R) p Fg 2680 4190 a(0) p Fo 2733 4227 a(dep) s(enden) m(t) p 3202 4227 a(only) p 3414 4227 a(on) p Fn 3547 4227 a(B) p Fo 3626 4227 a(,) p Fn 3684 4227 a(E) p Fo 3762 4227 a(,) p Fn 3819 4227 a(\016) p Fl 3862 4242 a(0) p Fo 0 4347 a(and) p Fn 190 4347 a(\013) p Fo 285 4347 a(suc) m(h) p 505 4347 a(that) p Fk 1474 4467 a(jj) p Fn(R) p Fm 1604 4482 a(N) p Fk 1671 4467 a(jj) p Fn 1754 4467 a(<) p Fo 1858 4467 a(1) p Fn(;) p Fo 1983 4467 a(if) p Fn 2079 4467 a(R) p Fk 2182 4467 a(\025) p Fn 2287 4467 a(R) p Fl 2361 4482 a(2) p Fn 2401 4467 a(:) p Fo 3653 4467 a(\(8.36\)) 0 4627 y(Since) p 254 4627 a(the) p 422 4627 a(\014rst) p 622 4627 a(term) p 855 4627 a(of) p 965 4627 a(the) p 1133 4627 a(righ) m(t) p 1368 4627 a(hand) p 1612 4627 a(side) p 1807 4627 a(of) p 1917 4627 a(\(8.35\)) p 2199 4627 a(is) p 2296 4627 a(a) p 2377 4627 a(linear) p 2648 4627 a(op) s(erator) p 3040 4627 a(of) p 3151 4627 a(at) p 3269 4627 a(most) p 3508 4627 a(rank) p Fn 3732 4627 a(N) p 3820 4627 a(k) p Fo 3874 4627 a(,) 0 4747 y(w) m(e) p 144 4747 a(ha) m(v) m(e) p 369 4747 a(\(8.27\)) p 650 4747 a(b) m(y) p 786 4747 a(Lemma) p 1134 4747 a(8.7) p 1291 4747 a(and) p 1481 4747 a(\(8.36\).) p 1758 4680 64 4 v 1758 4772 4 93 v 1817 4772 V 1758 4775 64 4 v 146 4936 a(Th) m(us) p 393 4936 a(the) p 561 4936 a(pro) s(of) p 816 4936 a(of) p 927 4936 a(Theorem) p 1339 4936 a(1.3) p 1496 4936 a(is) p 1594 4936 a(completed.) p 2098 4869 V 2098 4961 4 93 v 2158 4961 V 2098 4964 64 4 v 1902 5214 a(58) p eop %%Page: 59 59 59 58 bop Fh 0 631 a(References) p Fo 0 850 a([Ag]) p 639 850 a(Agmon,) p 1011 850 a(Shm) m(uel;) p Ff 1390 850 a(L) p 1446 850 a(e) p 1486 850 a(ctur) p 1651 850 a(es) p 1776 850 a(on) p 1920 850 a(el) p 1995 850 a(liptic) p 2246 850 a(b) p 2286 850 a(oundary) p 2670 850 a(value) p 2926 850 a(pr) p 3012 850 a(oblems) p Fo(,) p 3362 850 a(Prepared) p 3785 850 a(for) 639 970 y(publication) p 1149 970 a(b) m(y) p 1285 970 a(B.) p 1414 970 a(F) p 1470 970 a(rank) p 1695 970 a(Jones,) p 1990 970 a(Jr.) p 2138 970 a(with) p 2360 970 a(the) p 2529 970 a(assistance) p 2980 970 a(of) p 3092 970 a(George) p 3423 970 a(W.) p 3584 970 a(Batten,) 639 1091 y(Jr.) p 794 1091 a(V) p 859 1091 a(an) p 1002 1091 a(Nostrand) p 1435 1091 a(Mathematical) p 2063 1091 a(Studies,) p 2441 1091 a(No.) p 2630 1091 a(2) p 2718 1091 a(D.) p 2859 1091 a(V) p 2924 1091 a(an) p 3067 1091 a(Nostrand) p 3500 1091 a(Co.,) p 3715 1091 a(Inc.,) 639 1211 y(Princeton,) p 1112 1211 a(N.J.-T) p 1384 1211 a(oron) m(to-London,) p 2071 1211 a(1965.) 0 1414 y([Ah-Bo]) p 639 1414 a(Aharono) m(v,) p 1127 1414 a(Y.;) p 1303 1414 a(Bohm,) p 1630 1414 a(D.;) p 1807 1414 a(Signi\014cance) p 2349 1414 a(of) p 2471 1414 a(electromagnetic) p 3181 1414 a(p) s(oten) m(tials) p 3642 1414 a(in) p 3767 1414 a(the) 639 1535 y(quan) m(tum) p 1051 1535 a(theory) p 1316 1535 a(,) p Ff 1377 1535 a(Phys.) p 1644 1535 a(R) p 1710 1535 a(ev.) p Fq 1863 1535 a(115) p Fo 2063 1535 a(\(1959\)) p 2367 1535 a(485{491.) 0 1738 y([Al-Ge-Ho-Ho]) p 664 1738 a(Alb) s(ev) m(erio,) p 1164 1738 a(Sergio;) p 1524 1738 a(Gesztesy) p 1887 1738 a(,) p 1983 1738 a(F) p 2039 1738 a(riedric) m(h;) p 2461 1738 a(H\034egh-Krohn,) p 3126 1738 a(Raphael;) p 3574 1738 a(Holden,) 639 1858 y(Helge;) p Ff 966 1858 a(Solvable) p 1358 1858 a(mo) p 1483 1858 a(dels) p 1696 1858 a(in) p 1834 1858 a(quantum) p 2254 1858 a(me) p 2374 1858 a(chanics.) p 2771 1858 a(T) p 2834 1858 a(exts) p 3048 1858 a(and) p 3257 1858 a(Mono) p 3494 1858 a(gr) p 3575 1858 a(aphs) p 3817 1858 a(in) 639 1978 y(Physics.) p Fo(,) p 1047 1978 a(Springer-V) p 1503 1978 a(erlag,) p 1767 1978 a(New) p 1987 1978 a(Y) p 2052 1978 a(ork,) p 2250 1978 a(1988.) 0 2181 y([Ar1]) p 639 2181 a(Arai,) p 894 2181 a(Asao;) p 1173 2181 a(Momen) m(tum) p 1699 2181 a(op) s(erators) p 2137 2181 a(with) p 2366 2181 a(gauge) p 2649 2181 a(p) s(oten) m (tials,) p 3134 2181 a(lo) s(cal) p 3371 2181 a(quan) m(tization) 639 2302 y(of) p 756 2302 a(magnetic) p 1179 2302 a(\015ux,) p 1405 2302 a(and) p 1600 2302 a(represen) m(tation) p 2242 2302 a(of) p 2359 2302 a(canonical) p 2792 2302 a(comm) m(utation) p 3391 2302 a(relations,) p Ff 3821 2302 a(J.) 639 2422 y(Math.) p 924 2422 a(Phys.) p Fq 1189 2422 a(33) p Fo 1334 2422 a(\(1992\),) p 1664 2422 a(no.) p 1827 2422 a(10,) p 1984 2422 a(3374{3378.) 0 2625 y([Ar2]) p 639 2625 a(Arai,) p 908 2625 a(Asao;) p 1203 2625 a(Represen) m(tation-theoretic) p 2298 2625 a(asp) s(ects) p 2655 2625 a(of) p 2784 2625 a(t) m(w) m(o-dimensional) p 3523 2625 a(quan) m(tum) 639 2746 y(systems) p 995 2746 a(in) p 1102 2746 a(singular) p 1464 2746 a(v) m(ector) p 1750 2746 a(p) s(oten) m(tials:) p 2235 2746 a(canonical) p 2656 2746 a(comm) m(utation) p 3243 2746 a(relations,) p 3661 2746 a(quan-) 639 2866 y(tum) p 848 2866 a(algebras,) p 1258 2866 a(and) p 1451 2866 a(reduction) p 1887 2866 a(to) p 2009 2866 a(lattice) p 2310 2866 a(quan) m(tum) p 2724 2866 a(systems,) p Ff 3117 2866 a(J.) p 3235 2866 a(Math.) p 3522 2866 a(Phys.) p Fq 3790 2866 a(39) p Fo 639 2986 a(\(1998\),) p 970 2986 a(no.) p 1133 2986 a(5,) p 1241 2986 a(2476{2498.) 0 3189 y([Co-Ne]) p 639 3189 a(Cornean,) p 1065 3189 a(H.) p 1203 3189 a(D.;) p 1374 3189 a(Nenciu,) p 1737 3189 a(G.,) p 1908 3189 a(Tw) m (o-dimensional) p 2670 3189 a(magnetic) p 3094 3189 a(Sc) m(hr\177) p 3280 3189 a(odinger) p 3634 3189 a(op) s(era-) 639 3310 y(tors:) p 890 3310 a(width) p 1175 3310 a(of) p 1294 3310 a(mini) p 1525 3310 a(bands) p 1816 3310 a(in) p 1938 3310 a(the) p 2114 3310 a(tigh) m(t) p 2358 3310 a(binding) p 2719 3310 a(appro) m(ximation,) p Ff 3407 3310 a(A) n(nn.) p 3659 3310 a(Henri) 639 3430 y(Poinc) p 880 3430 a(ar) n(\023) p 972 3430 a(e) p Fq 1056 3430 a(1) p Fo 1144 3430 a(\(2000\),) p 1475 3430 a(no.) p 1637 3430 a(2,) p 1746 3430 a(203{222.) 0 3633 y([De]) p 639 3633 a(Deift,) p 906 3633 a(P) p 964 3633 a(.) p 1016 3633 a(A;) p 1141 3633 a(Applications) p 1700 3633 a(of) p 1803 3633 a(a) p 1877 3633 a(comm) m(utation) p 2462 3633 a(form) m(ula,) p Ff 2841 3633 a(Duke) p 3084 3633 a(Math.) p 3361 3633 a(J.) p Fq 3467 3633 a(45) p Fo 3604 3633 a(\(1978\),) 639 3754 y(no.) p 802 3754 a(2,) p 910 3754 a(267{310.) 0 3957 y([Ex-St-Vy]) p 639 3957 a(Exner,) p 967 3957 a(P) p 1025 3957 a(.;) 1131 3932 y(\024) 1129 3957 y(St'o) m(v) p 1334 3957 a(\023) p 1345 3957 a(\020) m(\024) p 1373 3957 a(cek,) p 1585 3957 a(P) p 1643 3957 a(.;) p 1747 3957 a(Vyt) p 1904 3957 a(\024) p 1910 3957 a(ras) p 2078 3957 a(,) p 2152 3957 a(P) p 2210 3957 a(.;) p 2314 3957 a(Generalized) p 2857 3957 a(b) s(oundary) p 3307 3957 a(conditions) p 3785 3957 a(for) 639 4077 y(the) p 811 4077 a(Aharono) m(v-Bohm) p 1547 4077 a(e\013ect) p 1808 4077 a(com) m(bined) p 2247 4077 a(with) p 2473 4077 a(a) p 2558 4077 a(homogeneous) p 3158 4077 a(magnetic) p 3578 4077 a(\014eld,) p Ff 3821 4077 a(J.) 639 4197 y(Math.) p 924 4197 a(Phys.) p Fq 1189 4197 a(43) p Fo 1334 4197 a(\(2002\),) p 1664 4197 a(no.) p 1827 4197 a(5,) p 1935 4197 a(2151{2168.) 0 4401 y([Ik-Ka]) p 639 4401 a(Ik) m(eb) s(e,) p 922 4401 a(T) p 984 4401 a(eruo;) p 1223 4401 a(Kato,) p 1489 4401 a(T) p 1551 4401 a(osio;) p 1770 4401 a(Uniqueness) p 2274 4401 a(of) p 2379 4401 a(the) p 2540 4401 a(self-adjoin) m(t) p 3036 4401 a(extension) p 3461 4401 a(of) p 3565 4401 a(singular) 639 4521 y(elliptic) p 959 4521 a(di\013eren) m(tial) p 1447 4521 a(op) s(erators,) p Ff 1905 4521 a(A) n(r) p 2011 4521 a(ch.) p 2171 4521 a(R) p 2237 4521 a(ational) p 2563 4521 a(Me) p 2690 4521 a(ch.) p 2850 4521 a(A) n(nal.) p Fq 3112 4521 a(9) p Fo 3200 4521 a(\(1962\),) p 3531 4521 a(77{92.) 0 4724 y([Iw]) p 639 4724 a(Iw) m(atsuk) p 966 4724 a(a,) p 1078 4724 a(Akira;) p 1378 4724 a(The) p 1579 4724 a(essen) m(tial) p 1969 4724 a(sp) s(ectrum) p 2397 4724 a(of) p 2509 4724 a(t) m(w) m(o-dimensional) p 3231 4724 a(Sc) m(hr\177) p 3417 4724 a(odinger) p 3767 4724 a(op-) 639 4844 y(erators) p 976 4844 a(with) p 1209 4844 a(p) s(erturb) s(ed) p 1674 4844 a(constan) m(t) p 2079 4844 a(magnetic) p 2506 4844 a(\014elds,) p Ff 2797 4844 a(J.) p 2922 4844 a(Math.) p 3216 4844 a(Kyoto) p 3515 4844 a(Univ.) p Fq 3790 4844 a(23) p Fo 639 4965 a(\(1983\),) p 970 4965 a(no.) p 1133 4965 a(3,) p 1241 4965 a(475{480.) 1902 5214 y(59) p eop %%Page: 60 60 60 59 bop Fo 0 631 a([Le]) p 639 631 a(Leinfelder,) p 1130 631 a(Herb) s(ert;) p 1532 631 a(Gauge) p 1844 631 a(in) m(v) p 1969 631 a(ariance) p 2312 631 a(of) p 2431 631 a(Sc) m(hr\177) p 2617 631 a(odinger) p 2973 631 a(op) s(erators) p 3412 631 a(and) p 3609 631 a(related) 639 751 y(sp) s(ectral) p 1006 751 a(prop) s(erties,) p Ff 1491 751 a(J.) p 1607 751 a(Op) p 1727 751 a(er) p 1808 751 a(ator) p 2016 751 a(The) p 2176 751 a(ory) p Fq 2355 751 a(9) p Fo 2443 751 a(\(1983\),) p 2774 751 a(no.) p 2936 751 a(1,) p 3045 751 a(163{179.) 0 955 y([Le-Si]) p 639 955 a(Leinfelder,) p 1118 955 a(Herb) s(ert;) p 1506 955 a(Simader,) p 1910 955 a(Christian) p 2336 955 a(G;) p 2469 955 a(Sc) m(hr\177) p 2655 955 a(odinger) p 2999 955 a(op) s(erators) p 3428 955 a(with) p 3647 955 a(singu-) 639 1075 y(lar) p 786 1075 a(magnetic) p 1203 1075 a(v) m(ector) p 1496 1075 a(p) s(oten) m(tials.) p Ff 1973 1075 a(Math.) p 2257 1075 a(Z.) p Fq 2380 1075 a(176) p Fo 2580 1075 a(\(1981\),) p 2911 1075 a(no.) p 3073 1075 a(1,) p 3182 1075 a(1{19.) 0 1279 y([Nak]) p 639 1279 a(Nak) p 807 1279 a(am) m(ura,) p 1154 1279 a(Sh) m(u;) p 1397 1279 a(A) p 1517 1279 a(remark) p 1866 1279 a(on) p 2016 1279 a(the) p 2200 1279 a(Diric) m(hlet-Neumann) p 3059 1279 a(decoupling) p 3562 1279 a(and) p 3767 1279 a(the) 639 1399 y(in) m(tegrated) p 1103 1399 a(densit) m(y) p 1439 1399 a(of) p 1551 1399 a(states,) p Ff 1855 1399 a(J.) p 1971 1399 a(F) p 2028 1399 a(unct.) p 2276 1399 a(A) n(nal.) p Fq 2538 1399 a(179) p Fo 2739 1399 a(\(2001\),) p 3069 1399 a(no.) p 3232 1399 a(1,) p 3340 1399 a(136{152.) 0 1602 y([Nam]) p 639 1602 a(Nam) m(bu,) p 1016 1602 a(Y) p 1081 1602 a(oic) m(hiro;) p 1434 1602 a(The) p 1641 1602 a(Aharono) m(v-Bohm) p 2380 1602 a(problem) p 2766 1602 a(revisited,) p Ff 3194 1602 a(Nucle) p 3428 1602 a(ar) p 3560 1602 a(Phys.) p 3833 1602 a(B) p Fq 639 1723 a(579) p Fo 840 1723 a(\(2000\),) p 1171 1723 a(no.) p 1333 1723 a(3,) p 1442 1723 a(590{616.) 0 1926 y([Re-Si]) p 639 1926 a(Reed,) p 918 1926 a(Mic) m(hael;) p 1316 1926 a(Simon,) p 1648 1926 a(Barry;) p Ff 1961 1926 a(Metho) p 2220 1926 a(ds) p 2349 1926 a(of) p 2469 1926 a(mo) p 2594 1926 a(dern) p 2823 1926 a(mathematic) p 3312 1926 a(al) p 3426 1926 a(physics.) p 3797 1926 a(II.) 639 2046 y(F) p 696 2046 a(ourier) p 990 2046 a(analysis,) p 1391 2046 a(self-adjointness) p Fo 2073 2046 a(Academic) p 2520 2046 a(Press,) p 2804 2046 a(New) p 3024 2046 a(Y) p 3089 2046 a(ork-London,) p 3640 2046 a(1975.) 0 2250 y([Ru1]) p 639 2250 a(Rudin,) p 961 2250 a(W) p 1053 2250 a(alter;) p Ff 1307 2250 a(R) p 1373 2250 a(e) p 1413 2250 a(al) p 1523 2250 a(and) p 1712 2250 a(c) p 1752 2250 a(omplex) p 2081 2250 a(analysis.) p 2482 2250 a(Thir) p 2668 2250 a(d) p 2752 2250 a(e) p 2792 2250 a(dition) p Fo(,) p 3098 2250 a(McGra) m(w-Hill) p 3681 2250 a(Bo) s(ok) 639 2370 y(Co.,) p 845 2370 a(New) p 1065 2370 a(Y) p 1130 2370 a(ork,) p 1328 2370 a(1987.) 0 2574 y([Ru2]) p 639 2574 a(Rudin,) p 974 2574 a(W) p 1066 2574 a(alter;) p Ff 1337 2574 a(F) p 1394 2574 a(unctional) p 1832 2574 a(analysis.) p 2243 2574 a(Se) p 2338 2574 a(c) p 2378 2574 a(ond) p 2577 2574 a(e) p 2617 2574 a(dition) p Fo(,) p 2937 2574 a(In) m(ternational) p 3533 2574 a(Series) p 3821 2574 a(in) 639 2694 y(Pure) p 874 2694 a(and) p 1064 2694 a(Applied) p 1429 2694 a(Mathematics,) p 2040 2694 a(McGra) m(w-Hill,) p 2650 2694 a(Inc.,) p 2869 2694 a(New) p 3089 2694 a(Y) p 3154 2694 a(ork,) p 3352 2694 a(1991.) 0 2897 y([W) p 119 2897 a(e]) p 639 2897 a(W) p 731 2897 a(eidmann,) p 1154 2897 a(Joac) m(him;) p Ff 1563 2897 a(Line) p 1749 2897 a(ar) p 1874 2897 a(op) p 1969 2897 a(er) p 2050 2897 a(ators) p 2297 2897 a(in) p 2415 2897 a(Hilb) p 2582 2897 a(ert) p 2735 2897 a(sp) p 2820 2897 a(ac) p 2910 2897 a(es) p Fo(,) p 3052 2897 a(T) p 3114 2897 a(ranslated) p 3537 2897 a(from) p 3767 2897 a(the) 639 3018 y(German) p 1030 3018 a(b) m(y) p 1181 3018 a(Joseph) p 1519 3018 a(Sz) s(\177) p 1617 3018 a(ucs,) p 1832 3018 a(Graduate) p 2282 3018 a(T) p 2344 3018 a(exts) p 2564 3018 a(in) p 2694 3018 a(Mathematics,) p Fq 3324 3018 a(68) p Fo(.) p 3512 3018 a(Springer-) 639 3138 y(V) p 704 3138 a(erlag,) p 970 3138 a(New) p 1189 3138 a(Y) p 1254 3138 a(ork-Berlin,) p 1744 3138 a(1980.) 1902 5214 y(60) p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0403120808222--