This is a multi-part message in MIME format. ---------------1112010659619 Content-Type: text/plain; name="11-184.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="11-184.comments" 15 pages ---------------1112010659619 Content-Type: text/plain; name="11-184.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="11-184.keywords" Unitary operators, spectral analysis, Mourre theory, cocycles over rotations ---------------1112010659619 Content-Type: application/postscript; name="unitary18.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="unitary18.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.96dev Copyright 2007 Radical Eye Software %%Title: C:/Documents and Settings/Rtiedra/Mis documentos/science/articles/articles_en_cours/Unitary_2012/unitary18.dvi %%CreationDate: Thu Dec 01 09:47:58 2011 %%Pages: 15 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: CMBX9 CMBX10 CMBX12 rsfs10 CMSS10 CMSS8 EUFM10 %%DocumentPaperSizes: Letter %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: %+ "C:\Archivos de programa\MiKTeX 2.7\miktex\bin\dvips.exe" %+ "C:/Documents and Settings/Rtiedra/Mis documentos/science/articles/articles_en_cours/Unitary_2012/unitary18.dvi" %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2011.12.01:0947 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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All Rights Reserved) readonly def /FullName (CMSS8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMSS8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 73 /I put dup 109 /m put readonly def /FontBBox{-65 -250 1062 761}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSS10 %!PS-AdobeFont-1.1: CMSS10 1.0 %%CreationDate: 1991 Aug 20 17:33:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSS10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMSS10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 73 /I put dup 76 /L put dup 82 /R put dup 97 /a put dup 99 /c put dup 101 /e put dup 108 /l put dup 109 /m put dup 110 /n put readonly def /FontBBox{-61 -250 999 759}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: rsfs10 %!PS-AdobeFont-1.0: rsfs10 001.000 %%CreationDate: Sat Mar 21 18:47:14 1998 %%VMusage: 120000 150000 11 dict begin /FontInfo 14 dict dup begin /version (001.001) readonly def /Copyright (Conversion from mf curves by Metafog (c) 1995 Richard Kinch) readonly def /Notice (Copyright (c) Taco Hoekwater, 1998. All rights reserved.) readonly def /FullName (rsfs10) readonly def /FamilyName (rsfs10) readonly def /ItalicAngle -12 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Roman) readonly def end readonly def /FontName /rsfs10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 66 /B put dup 75 /K put dup 83 /S put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox {-2 -300 1240 728} readonly def currentdict end currentfile eexec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CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 65 /A put dup 67 /C put dup 73 /I put dup 80 /P put dup 82 /R put dup 86 /V put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-53 -251 1139 750}readonly def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 67 /C put dup 69 /E put dup 75 /K put dup 76 /L put dup 77 /M put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-301 -250 1164 946}readonly def currentdict end currentfile eexec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(consider)f(the)j(closed)d(b)r(ounded)h(set)h Fr(J)k Fq(\032)23 b Fm(R)35 b Fw(giv)n(en)26 b(b)n(y)1406 3818 y Fr(J)31 b Fw(:=)1593 3701 y Fi(\032)1655 3818 y Fq(\000)p Fr(i)1769 3762 y Fw(1)18 b(+)1918 3743 y(\026)1912 3762 y Fr(\022)r(\022)1994 3732 y Fp(0)p 1769 3799 247 4 v 1769 3880 a Fw(1)g Fq(\000)1918 3860 y Fw(\026)1912 3880 y Fr(\022)r(\022)1994 3856 y Fp(0)2049 3818 y Fq(j)23 b Fr(\022)2136 3784 y Fp(0)2181 3818 y Fq(2)g Fw(\002)2324 3784 y Fp(0)2346 3701 y Fi(\033)2422 3818 y Fr(:)83 4041 y Fw(Since)k Fr(E)367 4011 y Fn(U)420 4041 y Fw(\(\002)517 4011 y Fp(0)539 4041 y Fw(\))c(=)g Fr(E)748 4011 y Fn(H)801 4019 y Fg(\022)836 4041 y Fw(\()p Fr(J)8 b Fw(\),)28 b(it)f(follo)n(ws)f(that)893 4214 y Fr(H)967 4180 y Fy(2)961 4235 y Fn(\022)1004 4214 y Fr(E)1070 4180 y Fn(U)1123 4214 y Fw(\(\002)1220 4180 y Fp(0)1242 4214 y Fw(\))d(=)g Fr(H)1459 4180 y Fy(2)1453 4235 y Fn(\022)1496 4214 y Fr(E)1562 4180 y Fn(H)1615 4188 y Fg(\022)1650 4214 y Fw(\()p Fr(J)8 b Fw(\))23 b(=)g Fr(\021)s Fw(\()p Fr(H)2024 4226 y Fn(\022)2060 4214 y Fw(\))p Fr(E)2158 4180 y Fn(H)2211 4188 y Fg(\022)2246 4214 y Fw(\()p Fr(J)8 b Fw(\))23 b(=)g Fr(\021)s Fw(\()p Fr(H)2620 4226 y Fn(\022)2656 4214 y Fw(\))p Fr(E)2754 4180 y Fn(U)2807 4214 y Fw(\(\002)2904 4180 y Fp(0)2926 4214 y Fw(\))83 4387 y(for)k(some)g(function)f Fr(\021)f Fq(2)f Fr(C)958 4357 y Fp(1)1028 4387 y Fw(\()p Fm(R)q Fw(;)14 b Fm(R)7 b Fw(\))34 b(with)27 b(compact)f(supp)r(ort.)g (As)i(a)f(consequence,)f(one)g(infers)h(that)303 4625 y Fr(\016)t Fw(\()p Fr(U)o(;)14 b(z)t Fw(\))p Fr(E)610 4591 y Fn(U)664 4625 y Fw(\(\002)761 4591 y Fp(0)783 4625 y Fw(\))23 b(=)926 4558 y Fi(\000)964 4625 y Fw(1)18 b Fq(\000)g(j)p Fr(z)t Fq(j)1197 4591 y Fy(2)1235 4558 y Fi(\001)1283 4569 y Fr(\021)s Fw(\()p Fr(H)1429 4581 y Fn(\022)1465 4569 y Fw(\))g(+)g(1)p 1283 4606 358 4 v 1306 4687 a Fq(j)p Fw(1)g Fq(\000)1478 4667 y Fw(\026)1472 4687 y Fr(\022)r(z)t Fq(j)1580 4663 y Fy(2)1664 4508 y Fi(\022)1725 4625 y Fr(H)1793 4637 y Fn(\022)1847 4625 y Fw(+)g Fr(i)1980 4569 y Fw(1)f(+)2129 4550 y(\026)2123 4569 y Fr(\022)q(z)p 1980 4606 229 4 v 1980 4687 a Fw(1)g Fq(\000)2129 4667 y Fw(\026)2123 4687 y Fr(\022)q(z)2218 4508 y Fi(\023)2279 4519 y Fp(\000)p Fy(1)2381 4508 y Fi(\022)2442 4625 y Fr(H)2510 4637 y Fn(\022)2564 4625 y Fw(+)p 2647 4489 288 4 v 18 w Fr(i)2696 4569 y Fw(1)h(+)2846 4550 y(\026)2839 4569 y Fr(\022)r(z)p 2696 4606 229 4 v 2696 4687 a Fw(1)g Fq(\000)2846 4667 y Fw(\026)2839 4687 y Fr(\022)r(z)2934 4508 y Fi(\023)2996 4526 y Fp(\000)p Fy(1)3084 4625 y Fr(E)3150 4591 y Fn(U)3203 4625 y Fw(\(\002)3300 4591 y Fp(0)3322 4625 y Fw(\))p Fr(;)220 b Fw(\(2.7\))83 4849 y(where)38 b Fr(\021)s Fw(\()p Fr(H)481 4861 y Fn(\022)517 4849 y Fw(\))k Fq(2)f Fr(C)752 4819 y Fy(1)790 4849 y Fw(\()p Fr(A)p Fw(\))e(if)g Fr(H)1110 4861 y Fn(\022)1184 4849 y Fw(is)f(of)h(class)e Fr(C)1655 4819 y Fy(1)1692 4849 y Fw(\()p Fr(A)p Fw(\),)j(due)e(to)g([3,)g(Thm.)g(6.2.5].)g(Under) f(the)i(same)e(regularit)n(y)83 4957 y(assumption)25 b(on)h Fr(H)707 4969 y Fn(\022)743 4957 y Fw(,)i(let)f(us)g(also)f (observ)n(e)g(that)i(for)f(an)n(y)g Fr(s)c Fq(2)g Fw([0)p Fr(;)14 b Fw(1\))27 b(the)h(op)r(erator)1054 5140 y Fq(h)p Fr(A)p Fq(i)1179 5105 y Fp(\000)p Fn(s)1266 5073 y Fi(\000)1304 5140 y Fr(\021)s Fw(\()p Fr(H)1450 5152 y Fn(\022)1486 5140 y Fw(\))19 b(+)f(1)1662 5073 y Fi(\001)1700 5140 y Fq(h)p Fr(A)p Fq(i)1825 5105 y Fn(s)1944 5140 y Fw(with)82 b Fq(h)p Fr(A)p Fq(i)24 b Fw(:=)2449 5058 y Fi(p)p 2532 5058 242 4 v 82 x Fw(1)18 b(+)g Fr(A)2736 5116 y Fy(2)2774 5140 y Fr(;)83 5313 y Fw(de\014ned)33 b(on)g Fq(D)565 5246 y Fi(\000)603 5313 y Fq(h)p Fr(A)p Fq(i)728 5283 y Fn(s)764 5246 y Fi(\001)802 5313 y Fw(,)h(extends)f(con)n(tin)n (uously)d(to)j(an)g(elemen)n(t)f(of)i Fj(B)s Fw(\()p Fq(H)q Fw(\))h(\(see)e([3,)h(Prop.)e(5.3.1]\).)h(With)h(these)83 5421 y(preparations)24 b(done,)i(w)n(e)i(can)e(pro)n(v)n(e)g(the)i (existence)e(of)i(a)f(large)f(class)g(of)i(lo)r(cally)d Fr(U)9 b Fw(-smo)r(oth)26 b(op)r(erators)9 b(:)p Black 1905 5670 a(7)p Black eop end %%Page: 8 8 TeXDict begin 8 7 bop Black Black Black 83 307 a Ft(Prop)s(osition)40 b(2.9)h Fw(\(Lo)r(cally)36 b Fr(U)9 b Fw(-smo)r(oth)33 b(op)r(erators\))p Ft(.)p Black 44 w Fx(Supp)-5 b(ose)41 b(that)e(the)g(assumptions)i(of)g(The)-5 b(or)g(em)41 b(2.7)83 415 y(hold)28 b(for)i(some)g(op)-5 b(en)30 b(set)g Fw(\002)23 b Fq(\032)f Fm(T)p Fx(.)31 b(Then,)f(e)-5 b(ach)30 b(op)-5 b(er)g(ator)28 b Fr(B)e Fq(2)e Fj(B)s Fw(\()p Fq(H)q Fw(\))31 b Fx(which)f(extends)g(c)-5 b(ontinuously)30 b(to)f(an)83 523 y(element)j(of)f Fj(B)596 456 y Fi(\000)635 523 y Fq(D)r Fw(\()p Fq(h)p Fr(A)p Fq(i)858 493 y Fn(s)895 523 y Fw(\))927 493 y Fp(\003)964 523 y Fr(;)14 b Fq(H)1072 456 y Fi(\001)1142 523 y Fx(for)31 b(some)i Fr(s)23 b(>)g Fw(1)p Fr(=)p Fw(2)30 b Fx(is)j(lo)-5 b(c)g(al)5 b(ly)30 b Fr(U)9 b Fx(-smo)-5 b(oth)31 b(on)h Fw(\002)18 b Fq(n)g Fr(\033)2904 535 y Fy(p)2944 523 y Fw(\()p Fr(U)9 b Fw(\))p Fx(.)p Black 83 691 a(Pr)-5 b(o)g(of.)p Black 42 w Fw(W)e(e)27 b(can)e(set)i Fr(B)g Fw(=)22 b Fq(h)p Fr(A)p Fq(i)1081 660 y Fp(\000)p Fn(s)1195 691 y Fw(and)j Fr(s)e Fq(2)h Fw(\(1)p Fr(=)p Fw(2)p Fr(;)14 b Fw(1\))25 b(without)h(lost)f(of)i (generalit)n(y)-7 b(.)23 b(So,)k(w)n(e)f(giv)n(e)g(the)g(pro)r(of)f(in) h(this)83 798 y(con)n(text.)249 906 y(Let)j(\002)463 876 y Fp(0)508 906 y Fm(\()22 b Fw(\002)d Fq(n)f Fr(\033)787 918 y Fy(p)828 906 y Fw(\()p Fr(U)9 b Fw(\))27 b(b)r(e)h(closed)e(and)g (\014x)j Fr(\022)c Fq(2)f Fw(\002)18 b Fq(n)g(f)p Fr(\033)2000 918 y Fy(p)2040 906 y Fw(\()p Fr(U)9 b Fw(\))19 b Fq([)g Fw(\002)2327 876 y Fp(0)2348 906 y Fq(g)p Fw(.)28 b(Let)g(also)e Fq(O)2823 876 y Fp(0)2869 906 y Fq(\032)d Fm(D)49 b Fw(b)r(e)28 b(an)f(op)r(en)f(set)i(with)83 1014 y(closure)d Fh(cl)p Fw(\()p Fq(O)518 984 y Fp(0)541 1014 y Fw(\))j(satisfying)1254 1122 y(\002)1319 1088 y Fp(0)1364 1122 y Fq(\032)22 b Fh(cl)p Fw(\()p Fq(O)1608 1088 y Fp(0)1631 1122 y Fw(\))c Fq(\\)h Fm(T)24 b Fq(\032)e Fw(\002)c Fq(n)2064 1055 y Fi(\000)2102 1122 y Fr(\033)2150 1134 y Fy(p)2191 1122 y Fw(\()p Fr(U)9 b Fw(\))18 b Fq([)h(f)p Fr(\022)r Fq(g)2537 1055 y Fi(\001)2574 1122 y Fr(:)83 1275 y Fw(Then,)27 b(one)f(infers)h(from)f(the)i(de\014nition)d(of)j Fr(\016)t Fw(\()p Fr(U)o(;)14 b(z)t Fw(\))28 b(that)1445 1457 y(sup)1394 1531 y Fn(z)s Fp(2)p Fd(D)p Fp(nO)1596 1514 y Ff(0)1635 1387 y Fi(\015)1635 1437 y(\015)1681 1457 y Fr(\016)t Fw(\()p Fr(U)o(;)14 b(z)t Fw(\))p Fr(E)1988 1423 y Fn(U)2043 1457 y Fw(\(\002)2140 1423 y Fp(0)2161 1457 y Fw(\))2193 1387 y Fi(\015)2193 1437 y(\015)2263 1457 y Fr(<)23 b Fq(1)p Fr(:)1140 b Fw(\(2.8\))83 1699 y(W)-7 b(e)28 b(can)f(th)n(us)f (restrict)h(our)f(atten)n(tion)g(to)h(the)h(case)f Fr(z)g Fq(2)c(O)1987 1669 y Fp(0)2010 1699 y Fw(.)28 b(But,)g(w)n(e)f(kno)n(w) g(from)g(\(2.7\))g(that)164 1881 y Fq(h)p Fr(A)p Fq(i)289 1847 y Fp(\000)p Fn(s)376 1881 y Fr(\016)t Fw(\()p Fr(U)o(;)14 b(z)t Fw(\))p Fr(E)683 1847 y Fn(U)737 1881 y Fw(\(\002)834 1847 y Fp(0)856 1881 y Fw(\))p Fq(h)p Fr(A)p Fq(i)1013 1847 y Fp(\000)p Fn(s)187 2090 y Fw(=)22 b Fq(h)p Fr(A)p Fq(i)399 2056 y Fp(\000)p Fn(s)486 2023 y Fi(\000)524 2090 y Fr(\021)s Fw(\()p Fr(H)670 2102 y Fn(\022)706 2090 y Fw(\))d(+)f(1)882 2023 y Fi(\001)920 2090 y Fq(h)p Fr(A)p Fq(i)1045 2056 y Fn(s)1080 1973 y Fi(\032)1173 2034 y Fw(1)g Fq(\000)g(j)p Fr(z)t Fq(j)1406 2004 y Fy(2)p 1153 2071 312 4 v 1153 2152 a Fq(j)p Fw(1)g Fq(\000)1325 2132 y Fw(\026)1319 2152 y Fr(\022)r(z)t Fq(j)1427 2128 y Fy(2)1474 2090 y Fq(h)p Fr(A)p Fq(i)1599 2056 y Fp(\000)p Fn(s)1700 1973 y Fi(\022)1761 2090 y Fr(H)1829 2102 y Fn(\022)1883 2090 y Fw(+)g Fr(i)2015 2034 y Fw(1)g(+)2165 2015 y(\026)2158 2034 y Fr(\022)r(z)p 2015 2071 229 4 v 2015 2152 a Fw(1)g Fq(\000)2165 2132 y Fw(\026)2158 2152 y Fr(\022)r(z)2253 1973 y Fi(\023)2315 1984 y Fp(\000)p Fy(1)2417 1973 y Fi(\022)2478 2090 y Fr(H)2546 2102 y Fn(\022)2600 2090 y Fw(+)p 2683 1954 288 4 v 18 w Fr(i)2732 2034 y Fw(1)g(+)2882 2015 y(\026)2875 2034 y Fr(\022)r(z)p 2732 2071 229 4 v 2732 2152 a Fw(1)g Fq(\000)2882 2132 y Fw(\026)2875 2152 y Fr(\022)r(z)2970 1973 y Fi(\023)3031 1991 y Fp(\000)p Fy(1)3120 2090 y Fr(E)3186 2056 y Fn(U)3239 2090 y Fw(\(\002)3336 2056 y Fp(0)3358 2090 y Fw(\))p Fq(h)p Fr(A)p Fq(i)3515 2056 y Fp(\000)p Fn(s)3602 1973 y Fi(\033)3664 2090 y Fr(;)83 2326 y Fw(for)36 b(some)g Fr(\021)k Fq(2)f Fr(C)679 2296 y Fp(1)749 2326 y Fw(\()p Fm(R)q Fw(;)14 b Fm(R)7 b Fw(\))43 b(with)36 b(compact)f(supp)r(ort.)g (F)-7 b(urthermore,)34 b(w)n(e)i(kno)n(w)g(that)h Fq(h)p Fr(A)p Fq(i)3075 2296 y Fp(\000)p Fn(s)3162 2259 y Fi(\000)3200 2326 y Fr(\021)s Fw(\()p Fr(H)3346 2338 y Fn(\022)3382 2326 y Fw(\))25 b(+)f(1)3570 2259 y Fi(\001)3607 2326 y Fq(h)p Fr(A)p Fq(i)3732 2296 y Fn(s)83 2434 y Fw(de\014ned)38 b(on)h Fq(D)576 2367 y Fi(\000)614 2434 y Fq(h)p Fr(A)p Fq(i)739 2404 y Fn(s)775 2367 y Fi(\001)853 2434 y Fw(extends)f(con)n (tin)n(uously)e(to)j(an)f(elemen)n(t)g(of)i Fj(B)s Fw(\()p Fq(H)q Fw(\).)g(So,)f(w)n(e)h(can)e(further)g(restrict)g(our)83 2542 y(atten)n(tion)26 b(to)h(the)h(op)r(erator)d(within)i(the)g(curly) g(brac)n(k)n(ets)f(for)h Fr(z)g Fq(2)c(O)2302 2512 y Fp(0)2325 2542 y Fw(.)249 2658 y(Using)i(the)i(notations)d Fr(")f Fw(:=)g Fh(Im)1264 2591 y Fi(\000)1319 2658 y Fq(\000)16 b Fr(i)1449 2626 y Fy(1+)1538 2612 y(\026)1533 2626 y Fn(\022)r(z)p 1449 2640 151 4 v 1449 2692 a Fy(1)p Fp(\000)1538 2678 y Fy(\026)1533 2692 y Fn(\022)r(z)1610 2591 y Fi(\001)1671 2658 y Fq(\021)1784 2618 y Fp(j)p Fn(z)s Fp(j)1857 2593 y Fe(2)1889 2618 y Fp(\000)p Fy(1)p 1768 2639 221 4 v 1768 2692 a Fp(j)p Fy(1)p Fp(\000)1876 2678 y Fy(\026)1871 2692 y Fn(\022)r(z)s Fp(j)1957 2676 y Fe(2)2026 2658 y Fw(and)25 b Fr(\025)f Fw(:=)e Fh(Re)2473 2591 y Fi(\000)2527 2658 y Fq(\000)16 b Fr(i)2658 2626 y Fy(1+)2747 2612 y(\026)2742 2626 y Fn(\022)r(z)p 2658 2640 151 4 v 2658 2692 a Fy(1)p Fp(\000)2747 2678 y Fy(\026)2742 2692 y Fn(\022)r(z)2818 2591 y Fi(\001)2880 2658 y Fq(\021)22 b(\000)3042 2618 y Fy(2)11 b Fc(Im)p Fy(\()p Fn(\022)6 b Fy(\026)-37 b Fn(z)s Fy(\))p 3042 2639 230 4 v 3046 2692 a Fp(j)p Fy(1)p Fp(\000)3154 2678 y Fy(\026)3149 2692 y Fn(\022)r(z)s Fp(j)3235 2676 y Fe(2)3281 2658 y Fw(,)26 b(one)g(gets)g(the)83 2766 y(follo)n(wing)f(b)r(ound)h(for)h (the)g(norm)f(the)i(op)r(erator)d(within)i(the)g(curly)f(brac)n(k)n (ets)9 b(:)242 2878 y Fi(\015)242 2928 y(\015)288 2949 y Fq(j)p Fr(")p Fq(jh)p Fr(A)p Fq(i)498 2914 y Fp(\000)p Fn(s)585 2881 y Fi(\000)623 2949 y Fr(H)691 2961 y Fn(\022)746 2949 y Fq(\000)18 b Fr(\025)h Fq(\000)f Fr(i")1048 2881 y Fi(\001)1086 2899 y Fp(\000)p Fy(1)1174 2881 y Fi(\000)1212 2949 y Fr(H)1280 2961 y Fn(\022)1335 2949 y Fq(\000)g Fr(\025)h Fw(+)f Fr(i")1637 2881 y Fi(\001)1675 2899 y Fp(\000)p Fy(1)1763 2949 y Fr(E)1829 2914 y Fn(U)1882 2949 y Fw(\(\002)1979 2914 y Fp(0)2001 2949 y Fw(\))p Fq(h)p Fr(A)p Fq(i)2158 2914 y Fp(\000)p Fn(s)2245 2878 y Fi(\015)2245 2928 y(\015)2315 2949 y Fq(\024)k(j)p Fr(")p Fq(j)2496 2878 y Fi(\015)2496 2928 y(\015)2543 2949 y Fq(h)p Fr(A)p Fq(i)2668 2914 y Fp(\000)p Fn(s)2755 2949 y Fw(\()p Fr(H)2855 2961 y Fn(\022)2909 2949 y Fq(\000)c Fr(\025)h Fq(\000)f Fr(i")p Fw(\))3243 2914 y Fp(\000)p Fy(1)3332 2878 y Fi(\015)3332 2928 y(\015)3378 2899 y Fy(2)3415 2949 y Fr(:)159 b Fw(\(2.9\))83 3143 y(No)n(w,)28 b(the)h(set)f Fq(V)j Fw(:=)763 3076 y Fi(\010)830 3143 y Fq(\000)18 b Fr(i)953 3110 y Fy(1+)1042 3096 y(\026)1037 3110 y Fn(\022)s(z)p 953 3124 151 4 v 953 3177 a Fy(1)p Fp(\000)1042 3163 y Fy(\026)1037 3177 y Fn(\022)s(z)1138 3143 y Fq(j)25 b Fr(z)j Fq(2)d Fh(cl)o Fw(\()p Fq(O)1490 3113 y Fp(0)1513 3143 y Fw(\))19 b Fq(\\)h Fm(T)1694 3076 y Fi(\011)1771 3143 y Fw(is)27 b(one)h(of)g(the)g(compact)f (subsets)g(of)h Fm(R)e Fq(n)18 b Fr(\033)3152 3155 y Fy(p)3193 3143 y Fw(\()p Fr(H)3293 3155 y Fn(\022)3329 3143 y Fw(\))29 b(men)n(tioned)83 3251 y(in)e(Prop)r(osition)d(2.6.)j (Hence,)g(one)g(infers)f(from)h(Prop)r(osition)d(2.6)j(that)557 3433 y(sup)544 3503 y Fn(z)s Fp(2O)677 3487 y Ff(0)710 3433 y Fq(j)p Fr(")p Fq(j)804 3363 y Fi(\015)804 3413 y(\015)851 3433 y Fq(h)p Fr(A)p Fq(i)976 3399 y Fp(\000)p Fn(s)1063 3433 y Fw(\()p Fr(H)1163 3445 y Fn(\022)1217 3433 y Fq(\000)18 b Fr(\025)h Fq(\000)f Fr(i")p Fw(\))1551 3399 y Fp(\000)p Fy(1)1640 3363 y Fi(\015)1640 3413 y(\015)1686 3383 y Fy(2)1746 3433 y Fw(=)36 b(sup)1834 3503 y Fn(z)s Fp(2O)1967 3487 y Ff(0)2000 3363 y Fi(\015)2000 3413 y(\015)2060 3433 y Fh(Im)2163 3366 y Fi(\010)2211 3433 y Fq(h)p Fr(A)p Fq(i)2336 3399 y Fp(\000)p Fn(s)2423 3433 y Fw(\()p Fr(H)2523 3445 y Fn(\022)2578 3433 y Fq(\000)18 b Fr(\025)h Fq(\000)f Fr(i")p Fw(\))2912 3399 y Fp(\000)p Fy(1)3000 3433 y Fq(h)p Fr(A)p Fq(i)3125 3399 y Fp(\000)p Fn(s)3212 3366 y Fi(\011)3261 3363 y(\015)3261 3413 y(\015)1746 3694 y Fq(\024)36 b Fw(sup)1834 3763 y Fn(z)s Fp(2O)1967 3747 y Ff(0)2000 3573 y Fi(\015)2000 3623 y(\015)2000 3673 y(\015)2000 3723 y(\015)2046 3694 y Fq(h)p Fr(A)p Fq(i)2171 3660 y Fp(\000)p Fn(s)2272 3577 y Fi(\022)2333 3694 y Fr(H)2401 3706 y Fn(\022)2455 3694 y Fw(+)18 b Fr(i)2587 3638 y Fw(1)g(+)2737 3618 y(\026)2730 3638 y Fr(\022)r(z)p 2587 3675 229 4 v 2587 3755 a Fw(1)g Fq(\000)2737 3736 y Fw(\026)2730 3755 y Fr(\022)r(z)2825 3577 y Fi(\023)2887 3588 y Fp(\000)p Fy(1)2989 3694 y Fq(h)p Fr(A)p Fq(i)3114 3660 y Fp(\000)p Fn(s)3200 3573 y Fi(\015)3200 3623 y(\015)3200 3673 y(\015)3200 3723 y(\015)1746 3886 y Fr(<)23 b Fq(1)p Fr(:)83 4068 y Fw(This)i(b)r(ound,) f(together)g(with)i(\(2.8\))f(and)g(\(2.9\),)h(implies)d(the)j(claim)e (since)g(an)n(y)h(closed)f(subset)h(of)h(\002)14 b Fq(n)h Fr(\033)3418 4080 y Fy(p)3458 4068 y Fw(\()p Fr(U)9 b Fw(\))26 b(is)f(at)83 4176 y(most)i(the)g(union)f(of)h(t)n(w)n(o)h (closed)d(sets)j(as)f(\002)1489 4146 y Fp(0)1510 4176 y Fw(.)p 3708 4176 4 57 v 3712 4124 50 4 v 3712 4176 V 3761 4176 4 57 v 249 4349 a(T)-7 b(o)23 b(close)f(the)i(section,)e(w) n(e)i(giv)n(e)e(a)h(corollary)e(on)i(the)g(p)r(erturbations)d(of)k Fr(U)31 b Fw(whic)n(h)23 b(is)g(useful)f(for)h(applications.)83 4457 y(F)-7 b(or)27 b(shortness,)e(w)n(e)j(state)f(the)h(result)e(in)h (the)g 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Fp(0)2072 1719 y Fw(\))p Fr(\021)s Fw(\()p Fr(V)k(U)9 b Fw(\),)27 b(it)h(follo)n(ws)e(that)100 1894 y Fr(E)166 1860 y Fn(V)14 b(U)271 1894 y Fw(\(\002)368 1860 y Fp(0)390 1894 y Fw(\))9 b(\()p Fr(V)19 b(U)9 b Fw(\))627 1860 y Fp(\003)664 1894 y Fw([)p Fr(A;)14 b(V)k(U)9 b Fw(])g Fr(E)1014 1860 y Fn(V)k(U)1119 1894 y Fw(\(\002)1216 1860 y Fp(0)1238 1894 y Fw(\))23 b(=)g Fr(E)1447 1860 y Fn(V)14 b(U)1552 1894 y Fw(\(\002)1649 1860 y Fp(0)1671 1894 y Fw(\))9 b Fr(\021)s Fw(\()p Fr(V)19 b(U)9 b Fw(\)\()p Fr(V)18 b(U)9 b Fw(\))2149 1860 y Fp(\003)2185 1894 y Fw([)p Fr(A;)14 b(V)19 b(U)9 b Fw(])g Fr(\021)s Fw(\()p Fr(V)17 b(U)9 b Fw(\))p Fr(E)2776 1860 y Fn(V)k(U)2881 1894 y Fw(\(\002)2978 1860 y Fp(0)3000 1894 y Fw(\))23 b Fq(\025)g Fr(a)9 b(E)3263 1860 y Fn(V)14 b(U)3369 1894 y Fw(\(\002)3466 1860 y Fp(0)3488 1894 y Fw(\))k(+)g Fr(K)3691 1906 y Fy(4)3728 1894 y Fr(;)83 2069 y Fw(with)28 b Fr(K)344 2081 y Fy(4)403 2069 y Fq(2)c Fj(K)i Fw(\()p Fq(H)q Fw(\).)i(Th)n(us,)e (the)i(claims)d(for)i 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Fr(A)p Fw(\))28 b(with)f Fr(U)3435 2910 y Fp(\003)3472 2940 y Fw([)p Fr(A;)14 b(U)9 b Fw(])22 b(=)83 3048 y(2)p Fr(T)12 b(P)251 3018 y Fy(2)286 3048 y Fw(\()p Fr(P)383 3018 y Fy(2)427 3048 y Fw(+)7 b(1\))573 3018 y Fp(\000)p Fy(1)660 3048 y Fw(,)22 b(and)f(further)f(computations)f(on)i Fj(S)16 b Fw(\()p Fm(R)1932 3018 y Fn(m)2002 3048 y Fw(\))22 b(sho)n(w)f(that)h Fr(U)31 b Fq(2)23 b Fr(C)2661 3018 y Fy(2)2699 3048 y Fw(\()p Fr(A)p Fw(\))h Fq(\032)f Fr(C)3001 3018 y Fy(1+0)3122 3048 y Fw(\()p Fr(A)p Fw(\).)g(No)n(w,)f(observ)n(e) 83 3156 y(that)j(for)f(an)n(y)g(op)r(en)g(set)g(\002)f Fq(\032)g Fm(T)i Fw(with)g(closure)d Fh(cl)p Fw(\(\002\))j(satisfying)e Fh(cl)p Fw(\(\002\))13 b Fq(\\)g(f)p Fw(1)p Fq(g)21 b Fw(=)i Fm(?)p Fw(,)h(there)h(exists)f Fr(\016)j(>)c Fw(0)h(suc)n(h)f (that)1402 3345 y Fr(E)1468 3311 y Fn(U)1521 3345 y Fw(\(\002\))g(=)g Fr(E)1827 3311 y Fn(P)1877 3286 y Fe(2)1913 3278 y Fi(\000)1951 3345 y Fw([)p Fr(\016;)14 b Fq(1)p Fw(\))2163 3278 y Fi(\001)2201 3345 y 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b(\022)r Fw(\),)28 b(see)f(for)g(example)f ([14)o(,)i(15)o(,)f(17]\).)249 1653 y(In)22 b(the)g(sequel,)f(w)n(e)h (treat)f(the)h(case)f Fr(f)32 b Fw(:=)23 b Fr(m)14 b Fw(id)f(+)p Fr(h)p Fw(,)22 b(with)g Fr(m)h Fq(2)h Fm(Z)2310 1623 y Fp(\003)2341 1653 y Fw(,)e(id)h(:)g([0)p Fr(;)14 b Fw(1\))22 b Fq(!)h Fw([0)p Fr(;)14 b Fw(1\))22 b(the)f(iden)n(tit)n (y)g(function)83 1761 y(and)j Fr(h)f Fw(:)h([0)p Fr(;)14 b Fw(1\))22 b Fq(!)h Fm(R)32 b Fw(an)24 b(absolutely)f(con)n(tin)n (uous)e(function)j(satisfying)f Fr(h)p Fw(\(0\))h(=)e Fr(h)p Fw(\(1\).)k(F)-7 b(or)24 b(this,)h(w)n(e)g(in)n(tro)r(duce)d (the)83 1869 y(self-adjoin)n(t)j(op)r(erator)g Fr(P)39 b Fw(in)27 b Fq(H)h Fw(de\014ned)f(b)n(y)132 2043 y Fr(P)12 b(')22 b Fw(:=)h Fq(\000)p Fr(i')535 2009 y Fp(0)556 2043 y Fr(;)97 b(')22 b Fq(2)i(D)r Fw(\()p Fr(P)12 b Fw(\))23 b(:=)1161 1976 y Fi(\010)1209 2043 y Fr(')g Fq(2)g(H)h(j)f Fr(')k Fw(is)g(absolutely)f(con)n(tin)n(uous)l Fr(;)41 b(')2577 2009 y Fp(0)2622 2043 y Fq(2)23 b Fh(L)2738 2009 y Fy(2)2775 1976 y Fi(\000)2813 2043 y Fw([0)p Fr(;)14 b Fw(1\))2989 1976 y Fi(\001)3054 2043 y Fw(and)27 b Fr(')p Fw(\(0\))22 b(=)h Fr(')p Fw(\(1\))3648 1976 y Fi(\011)3696 2043 y Fr(:)83 2218 y Fw(Then,)k(w)n(e)g(observ)n(e)f (that)i(an)f(in)n(tegration)d(b)n(y)j(parts)f(giv)n(es)h(for)g(eac)n(h) f Fr(')d Fq(2)g(D)r Fw(\()p Fr(P)12 b Fw(\))28 b(that)1102 2325 y Fi(\012)1141 2392 y Fr(P)12 b(';)i(U)9 b(')1418 2325 y Fi(\013)1473 2392 y Fq(\000)1556 2325 y Fi(\012)1595 2392 y Fr(';)14 b(U)9 b(P)j(')1872 2325 y Fi(\013)1932 2392 y Fw(=)2020 2325 y Fi(\012)2059 2392 y Fr(';)i Fw(2)p Fr(\031)2245 2325 y Fi(\000)2282 2392 y Fr(m)19 b Fw(+)f Fr(h)2508 2358 y Fp(0)2530 2325 y Fi(\001)2568 2392 y Fr(U)9 b(')2688 2325 y Fi(\013)2726 2392 y Fr(;)83 2567 y Fw(with)29 b Fr(h)324 2537 y Fp(0)376 2567 y Fw(the)g(op)r(erator)e (of)j(m)n(ultiplication)24 b(b)n(y)29 b Fr(h)1658 2537 y Fp(0)1681 2567 y Fw(.)g(Therefore,)f(if)h Fr(h)2262 2537 y Fp(0)2310 2567 y Fq(2)e Fh(L)2430 2537 y Fp(1)2499 2500 y Fi(\000)2537 2567 y Fw([0)p Fr(;)14 b Fw(1\))2713 2500 y Fi(\001)2750 2567 y Fw(,)30 b(one)e(infers)g(that)h Fr(U)34 b Fq(2)27 b Fr(C)3602 2537 y Fy(1)3639 2567 y Fw(\()p Fr(P)12 b Fw(\))83 2675 y(with)28 b Fr(U)339 2645 y Fp(\003)375 2675 y Fw([)p Fr(P)r(;)14 b(U)9 b Fw(])23 b(=)f(2)p Fr(\031)782 2608 y Fi(\000)820 2675 y Fr(m)d Fw(+)f Fr(h)1046 2645 y Fp(0)1068 2675 y Fw(\([)c Fq(\001)28 b(\000)18 b Fr(\022)r Fw(]\))1367 2608 y Fi(\001)1406 2675 y Fw(.)27 b(Moreo)n(v)n(er,)f(b)n(y)h(imp)r(osing)e(some)h (additional)e(condition)h(on)i(the)g(size)g(of)83 2783 y Fr(h)132 2753 y Fp(0)154 2783 y Fw(,)j(one)f(w)n(ould)e(also)h (obtain)g(that)h Fr(U)1282 2753 y Fp(\003)1319 2783 y Fw([)p Fr(P)r(;)14 b(U)9 b Fw(])29 b(has)f(a)h(de\014nite)g(sign)f (\(the)h(one)g(of)g Fr(m)p Fw(\).)i(Ho)n(w)n(ev)n(er,)d(since)g(this)h (is)g(not)83 2891 y(v)n(ery)23 b(satisfactory)-7 b(,)21 b(w)n(e)i(shall)e(explain)g(in)i(the)f(next)i(prop)r(osition)19 b(ho)n(w)j(this)g(additional)e(condition)g(can)i(b)r(e)h(a)n(v)n(oided) 83 2999 y(b)n(y)h(mo)r(difying)f(adequately)h(the)g(conjugate)f(op)r (erator)g Fr(P)12 b Fw(.)24 b(The)g(tric)n(k)g(is)h(based)e(on)h(the)h (ergo)r(dic)d(theorem)i(and)f(the)83 3107 y(new)28 b(conjugate)d(op)r (erator)g(is)i(of)h(the)f(form)1540 3347 y Fr(P)1593 3359 y Fn(n)1662 3347 y Fw(:=)1788 3291 y(1)p 1782 3328 53 4 v 1782 3404 a Fr(n)1859 3243 y Fn(n)p Fp(\000)p Fy(1)1862 3268 y Fi(X)1864 3447 y Fn(j)s Fy(=0)1998 3347 y Fr(U)2063 3313 y Fp(\000)p Fn(j)2148 3347 y Fr(P)12 b(U)2278 3313 y Fn(j)3597 3347 y Fw(\(3.3\))83 3604 y(for)35 b(some)g(large)e Fr(n)k Fq(2)g Fm(N)880 3574 y Fp(\003)958 3604 y Fw(\(see)f(the)f(App)r(endix)g(for)g(abstract)f(results)g(on)h (the)g(self-adjoin)n(tness)c(of)36 b Fr(P)3429 3616 y Fn(n)3511 3604 y Fw(and)e(on)83 3712 y(regularit)n(y)25 b(prop)r(erties)g(with)i(resp)r(ect)g(to)g Fr(P)1484 3724 y Fn(n)1530 3712 y Fw(\).)p Black 83 3873 a Ft(Prop)s(osition)35 b(3.2.)p Black 42 w Fx(L)-5 b(et)35 b Fr(f)j Fw(:=)29 b Fr(m)14 b Fw(id)g(+)p Fr(h)p Fx(,)35 b(with)h Fr(m)30 b Fq(2)g Fm(Z)1948 3843 y Fp(\003)2015 3873 y Fx(and)35 b Fr(h)30 b Fq(2)g Fr(C)2421 3843 y Fy(1)2458 3806 y Fi(\000)2496 3873 y Fw([0)p Fr(;)14 b Fw(1\);)g Fm(R)2763 3806 y Fi(\001)2842 3873 y Fx(with)36 b Fr(h)3089 3843 y Fp(0)3147 3873 y Fx(Dini-c)-5 b(ontinuous)83 3981 y(and)33 b Fr(h)p Fw(\(0\))25 b(=)f Fr(h)p Fw(\(1\))p Fx(.)34 b(Then)f(the)f(op)-5 b(er)g(ator)31 b Fr(U)9 b Fx(,)33 b(de\014ne)-5 b(d)33 b(by)41 b Fw(\(3.2\))32 b Fx(with)h Fr(\022)i Fx(irr)-5 b(ational,)31 b(has)i(a)g(pur)-5 b(ely)32 b(L)-5 b(eb)g(esgue)83 4089 y(sp)g(e)g(ctrum)32 b(e)-5 b(qual)32 b(to)f Fm(T)p Fx(.)h(F)-7 b(urthermor)i(e,)31 b(the)g(op)-5 b(er)g(ator)30 b Fq(h)p Fr(P)12 b Fq(i)2076 4059 y Fp(\000)p Fn(s)2193 4089 y Fx(is)33 b(glob)-5 b(al)5 b(ly)30 b Fr(U)9 b Fx(-smo)-5 b(oth)31 b(for)g(any)h Fr(s)24 b(>)e Fw(1)p Fr(=)p Fw(2)p Fx(.)p Black 83 4250 a(Pr)-5 b(o)g(of.)p Black 42 w Fw(In)29 b(what)g(follo)n(ws,)f(w)n(e)h (assume)e(without)i(loss)f(of)h(generalit)n(y)d(that)j Fr(m)e Fq(2)f Fm(N)2756 4220 y Fp(\003)2828 4250 y Fw(\(if)k Fr(m)c(<)f Fw(0,)k(the)g(same)f(pro)r(of)83 4358 y(w)n(orks)f(with)g Fr(P)39 b Fw(replaced)26 b(b)n(y)h Fq(\000)p Fr(P)12 b Fw(\).)249 4466 y(Since)28 b Fr(h)517 4436 y Fp(0)569 4466 y Fw(is)g(Dini-con)n(tin)n(uous,)c Fr(h)1330 4436 y Fp(0)1378 4466 y Fq(2)i Fh(L)1497 4436 y Fp(1)1566 4399 y Fi(\000)1604 4466 y Fw([0)p Fr(;)14 b Fw(1\))1780 4399 y Fi(\001)1847 4466 y Fw(and)28 b Fr(U)33 b Fq(2)26 b Fr(C)2245 4436 y Fy(1)2283 4466 y Fw(\()p Fr(P)12 b Fw(\))29 b(as)f(indicated)f(b)r(efore.)h(Th)n(us,)g(one)g(infers)83 4574 y(from)c(the)g(App)r(endix)g(\(with)h Fr(A)e Fw(=)g Fr(P)12 b Fw(\))24 b(that)g(for)h(eac)n(h)e Fr(n)g Fq(2)h Fm(N)2002 4544 y Fp(\003)2070 4574 y Fw(the)g(op)r(erator)f Fr(P)2595 4586 y Fn(n)2641 4574 y Fw(,)h(de\014ned)g(in)g(\(3.3\))o(,)h (is)f(self-adjoin)n(t)83 4682 y(on)j Fq(D)r Fw(\()p Fr(P)350 4694 y Fn(n)396 4682 y Fw(\))d(:=)e Fq(D)r Fw(\()p Fr(P)12 b Fw(\).)28 b(Moreo)n(v)n(er,)e(one)h(deduces)e(from)i(Lemma)g (4.1\(a\))g(that)g Fr(U)36 b Fw(b)r(elongs)26 b(to)h Fr(C)3128 4652 y Fy(1)3166 4682 y Fw(\()p Fr(P)3251 4694 y Fn(n)3297 4682 y Fw(\))h(with)276 4927 y([)p Fr(P)352 4939 y Fn(n)398 4927 y Fr(;)14 b(U)9 b Fw(])22 b(=)648 4871 y(1)p 643 4908 V 643 4984 a Fr(n)719 4823 y Fn(n)p Fp(\000)p Fy(1)722 4848 y Fi(X)725 5027 y Fn(j)s Fy(=0)859 4927 y Fr(U)924 4893 y Fp(\000)p Fn(j)1009 4927 y Fw([)p Fr(P)r(;)14 b(U)9 b Fw(])p Fr(U)1277 4893 y Fn(j)1333 4927 y Fw(=)1431 4871 y(2)p Fr(\031)p 1431 4908 94 4 v 1451 4984 a(n)1548 4823 y Fn(n)p Fp(\000)p Fy(1)1551 4848 y Fi(X)1554 5027 y Fn(j)s Fy(=0)1688 4927 y Fr(U)1753 4893 y Fp(\000)p Fn(j)1838 4860 y Fi(\000)1876 4927 y Fr(m)18 b Fw(+)h Fr(h)2102 4893 y Fp(0)2124 4860 y Fi(\001)2162 4927 y Fr(U)2227 4893 y Fn(j)s Fy(+1)2368 4927 y Fw(=)k(2)p Fr(\031)s(U)2615 4785 y Fi( )2679 4927 y Fr(m)c Fw(+)2871 4871 y(1)p 2866 4908 53 4 v 2866 4984 a Fr(n)2981 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2330 5294 V 2330 5342 a Fn(n)2396 5251 y Fi(P)2484 5271 y Fn(n)2484 5338 y(j)s Fy(=1)2616 5313 y Fr(h)2665 5283 y Fp(0)2688 5313 y Fw(\([)14 b Fq(\001)45 b(\000)23 b Fr(j)5 b(\022)r Fw(]\))3049 5243 y Fi(\014)3049 5292 y(\014)3112 5313 y Fr(<)35 b Fw(1)p Fr(=)p Fw(2)e(for)i Fr(n)g Fw(large)83 5421 y(enough.)25 b(So,)j(one)e(deduces)g(that)i Fr(U)1240 5391 y Fp(\003)1276 5354 y Fi(\002)1311 5421 y Fr(P)1364 5433 y Fn(n)1410 5421 y Fr(;)14 b(U)1512 5354 y Fi(\003)1569 5421 y Fq(\025)22 b Fr(\031)31 b Fw(if)c Fr(n)h Fw(is)f(large)f(enough.)p Black 1884 5670 a(11)p Black eop end %%Page: 12 12 TeXDict begin 12 11 bop Black Black 249 307 a Fw(T)-7 b(o)35 b(c)n(hec)n(k)g(that)h Fr(U)44 b Fq(2)37 b Fr(C)1057 277 y Fy(1+0)1179 307 y Fw(\()p Fr(P)1264 319 y Fn(n)1310 307 y Fw(\),)f(w)n(e)g(observ)n(e)e(from)h(Lemma)g(4.1\(b\))g(that)g (it)h(is)f(su\016cien)n(t)f(to)i(sho)n(w)e(the)83 415 y(inclusion)d Fr(U)43 b Fq(2)35 b Fr(C)693 385 y Fy(1+0)814 415 y Fw(\()p Fr(P)12 b Fw(\).)35 b(W)-7 b(e)35 b(also)e(recall)f(that) j([)p Fr(P)r(;)14 b(U)9 b Fw(])34 b(=)g(2)p Fr(\031)2168 348 y Fi(\000)2205 415 y Fr(m)24 b Fw(+)f Fr(h)2441 385 y Fp(0)2463 348 y Fi(\001)2501 415 y Fr(U)9 b Fw(.)34 b(Therefore,)f(since)g Fr(U)43 b Fw(satis\014es)33 b(the)83 536 y(condition)450 469 y Fi(R)505 490 y Fy(1)489 566 y(0)566 504 y(d)p Fn(t)p 566 518 62 4 v 585 565 a(t)647 466 y Fi(\015)647 516 y(\015)707 534 y Fw(e)744 506 y Fp(\000)p Fn(itP)911 536 y Fr(U)989 534 y Fw(e)1026 506 y Fn(itP)1142 536 y Fq(\000)p Fr(U)1272 466 y Fi(\015)1272 516 y(\015)1340 536 y Fr(<)23 b Fq(1)p Fw(,)28 b(one)f(is)f(reduced)g (to)i(sho)n(wing)d(that)602 651 y Fi(Z)685 672 y Fy(1)648 840 y(0)746 708 y Fw(d)p Fr(t)p 746 745 77 4 v 770 821 a(t)842 694 y Fi(\015)842 744 y(\015)902 762 y Fw(e)939 730 y Fp(\000)p Fn(itP)1093 764 y Fw(\()p Fr(m)19 b Fw(+)f Fr(h)1351 730 y Fp(0)1373 764 y Fw(\))1419 762 y(e)1456 730 y Fn(itP)1573 764 y Fq(\000)p Fw(\()p Fr(m)h Fw(+)f Fr(h)1896 730 y Fp(0)1918 764 y Fw(\))1950 694 y Fi(\015)1950 744 y(\015)2019 764 y Fw(=)2107 651 y Fi(Z)2190 672 y Fy(1)2153 840 y(0)2251 708 y Fw(d)p Fr(t)p 2251 745 V 2274 821 a(t)2347 694 y Fi(\015)2347 744 y(\015)2407 762 y Fw(e)2444 730 y Fp(\000)p Fn(itP)2611 764 y Fr(h)2660 730 y Fp(0)2696 762 y Fw(e)2733 730 y Fn(itP)2850 764 y Fq(\000)p Fr(h)2964 730 y Fp(0)2986 694 y Fi(\015)2986 744 y(\015)3055 764 y Fr(<)23 b Fq(1)p Fr(:)83 966 y Fw(Ho)n(w)n(ev)n(er,)j(this)g(condition)e(is)i(readily)e(v)n(eri\014ed) i(since)f(it)i(corresp)r(onds)c(to)k(nothing)d(else)i(but)g(the)h (Dini-con)n(tin)n(uit)n(y)83 1074 y(of)j Fr(h)229 1044 y Fp(0)251 1074 y Fw(.)g(One)f(then)g(concludes)e(b)n(y)j(applying)d (Theorem)h(2.7)h(and)f(Prop)r(osition)f(2.9,)i(and)f(b)n(y)i(taking)f (in)n(to)f(accoun)n(t)83 1182 y(the)f(equalit)n(y)g Fq(D)r Fw(\()p Fr(P)695 1194 y Fn(n)741 1182 y Fw(\))d(=)e Fq(D)r Fw(\()p Fr(P)12 b Fw(\).)p 3708 1182 4 57 v 3712 1130 50 4 v 3712 1182 V 3761 1182 4 57 v 249 1352 a(The)18 b(\014rst)h(part)e(of)i(Prop)r(osition)c(3.2)j(is)g(not)g(new;)h(the)f (nature)f(of)i(the)f(sp)r(ectrum)f(of)i Fr(U)27 b Fw(w)n(as)18 b(already)f(determined)83 1460 y(in)24 b([14)o(,)h(15)o(])g(under)e(a)h (sligh)n(tly)f(w)n(eak)n(er)h(assumption)e(\()p Fr(h)j Fw(absolutely)e(con)n(tin)n(uous)e(with)j Fr(h)2914 1430 y Fp(0)2962 1460 y Fw(of)g(b)r(ounded)f(v)-5 b(ariation\).)83 1568 y(On)34 b(the)g(other)f(hand,)g(w)n(e)h(ha)n(v)n(e)f(not)h(b)r (een)g(able)f(to)h(\014nd)g(in)f(the)h(literature)e(an)n(y)i (information)d(ab)r(out)i(globally)83 1676 y Fr(U)9 b Fw(-smo)r(oth)25 b(op)r(erators.)g(Therefore,)h(the)i(second)e(part)g (Prop)r(osition)f(3.2)h(is)h(apparen)n(tly)e(new.)83 1928 y Fl(3.4)112 b(V)-9 b(ector)36 b(\014elds)j(on)e(manifolds)83 2093 y Fw(Let)22 b Fr(M)30 b Fw(b)r(e)21 b(a)g(smo)r(oth)g(orien)n (table)d(manifold)g(of)k(dimension)c Fr(n)23 b Fq(\025)g Fw(1)e(with)g(v)n(olume)f(form)g(\012,)i(and)e(let)h Fq(H)j Fw(:=)f Fh(L)3487 2063 y Fy(2)3524 2093 y Fw(\()p Fr(M)t(;)14 b 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Fy(0)806 3219 y Fr(;)14 b Fw(\012\))26 b(=)g(1)j(and)f Fr(U)1342 3231 y Fn(t)1396 3219 y Fa(\026)e Fh(L)1495 3189 y Fy(2)1532 3219 y Fw(\()p Fr(M)i Fq(n)19 b Fr(M)1811 3231 y Fy(0)1849 3219 y Fr(;)14 b Fw(\012\))29 b(unitary)-7 b(.)28 b(Therefore,)g(w)n(e) h(can)g(restrict)f(our)g(atten)n(tion)83 3327 y(to)f(the)h(unitary)e (op)r(erator)f Fr(U)1021 3297 y Fy(0)1012 3348 y Fn(t)1081 3327 y Fw(:=)e Fr(U)1248 3339 y Fn(t)1299 3327 y Fa(\026)g Fh(L)1395 3297 y Fy(2)1431 3327 y Fw(\()p Fr(M)28 b Fq(n)18 b Fr(M)1709 3339 y Fy(0)1746 3327 y Fr(;)c Fw(\012\))28 b(in)f Fq(H)2071 3339 y Fy(0)2131 3327 y Fw(:=)c Fh(L)2280 3297 y Fy(2)2317 3327 y Fw(\()p Fr(M)k Fq(n)18 b Fr(M)2594 3339 y Fy(0)2631 3327 y Fr(;)c Fw(\012\).)249 3435 y(Supp)r(ose)26 b(that)i(there)e(exist)i(a)f(function)f Fr(g)g Fq(2)e Fr(C)1771 3405 y Fy(1)1808 3435 y Fw(\()p Fr(M)k Fq(n)18 b Fr(M)2086 3447 y Fy(0)2123 3435 y Fw(;)c Fm(R)q Fw(\))34 b(and)26 b(a)i(constan)n(t)d Fr(\016)i(>)c Fw(0)k(suc)n(h)f(that)1055 3603 y(d)p 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Fr(;)14 b Fq(1)p Fw(\);)g Fm(R)970 1333 y Fi(\001)1046 1401 y Fw(satis\014es)30 b(lim)1471 1413 y Fn(r)r Fp(&)p Fy(0)1621 1401 y Fq(j)p Fr(f)8 b Fw(\()p Fr(r)r Fw(\))p Fr(=r)r Fq(j)30 b Fr(<)e Fq(1)j Fw(and)f Fr(\016)2343 1413 y Fy(1)2381 1401 y Fr(r)h Fq(\024)d Fr(f)8 b Fw(\()p Fr(r)r Fw(\))30 b Fq(\024)f Fr(\016)2859 1413 y Fy(2)2896 1401 y Fr(r)23 b Fw(+)d Fr(\016)3080 1413 y Fy(3)3149 1401 y Fw(for)30 b(all)g Fr(r)h Fq(\025)e Fw(0)h(and)83 1509 y(some)d Fr(\016)329 1521 y Fy(1)366 1509 y Fr(;)14 b(\016)440 1521 y Fy(2)477 1509 y Fr(;)g(\016)551 1521 y Fy(3)612 1509 y Fr(>)24 b Fw(0.)k(Then,)f(w)n(e)h(ha)n(v)n(e)f Fr(M)1429 1521 y Fy(0)1490 1509 y Fw(=)d Fq(f)p Fw(0)p Fq(g)p Fw(,)j(and)g(w)n(e)h(kno)n(w)g(from)f([18)o(,)h(Rem.)g(1.8.7])f (that)h Fr(X)34 b Fw(is)28 b(complete.)83 1617 y(Therefore,)35 b(the)h(restriction)e Fr(U)1117 1586 y Fy(0)1108 1637 y Fn(t)1192 1617 y Fw(:=)j Fr(U)1373 1629 y Fn(t)1439 1617 y Fa(\026)h Fh(L)1550 1586 y Fy(2)1587 1617 y Fw(\()p Fm(R)1673 1586 y 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1952 y Fn(t)2117 1940 y Fw(is)g(purely)e(absolutely)g(con)n(tin)n(uous) e(in)j Fh(L)3406 1910 y Fy(2)3443 1940 y Fw(\()p Fm(R)3530 1910 y Fn(n)3582 1940 y Fr(;)14 b Fw(d)p Fr(x)p Fw(\).)83 2048 y(This)27 b(co)n(v)n(ers)e(for)i(instance)f(the)i(w)n(ell)e(kno)n (wn)h(case)g(of)h(the)f(dilation)e(group,)g(where)i Fr(f)8 b Fw(\()p Fr(r)r Fw(\))25 b(=)d Fr(r)r Fw(.)83 2331 y Fs(4)135 b(App)t(endix)83 2526 y Fw(In)30 b(this)f(app)r(endix,)e(w)n (e)j(in)n(tro)r(duce)e(an)h(abstract)f(class)h(of)g(conjugate)f(op)r (erators)f(whic)n(h)i(is)g(useful)g(for)g(the)h(study)83 2634 y(of)e(co)r(cycles)e(in)h(Section)f(3.3.)249 2742 y(Let)33 b Fr(A)f Fw(and)f Fr(U)40 b Fw(b)r(e)32 b(resp)r(ectiv)n(ely)e (a)i(self-adjoin)n(t)c(and)j(a)h(unitary)e(op)r(erator)g(in)h(a)g(Hilb) r(ert)g(space)g Fq(H)q Fw(.)h(If)g Fr(U)39 b Fq(2)83 2850 y Fr(C)148 2820 y Fy(1)186 2850 y Fw(\()p Fr(A)p Fw(\),)24 b(then)f(w)n(e)h(kno)n(w)g(from)e([3,)i(Prop.)e(5.1.5-5.1.6]) f(that)j Fr(U)2062 2820 y Fn(k)2124 2850 y Fq(2)g Fr(C)2268 2820 y Fy(1)2305 2850 y Fw(\()p Fr(A)p Fw(\))h(for)e(eac)n(h)g Fr(k)h Fq(2)g Fm(Z)18 b Fw(and)k Fr(U)3210 2820 y Fn(k)3250 2850 y Fq(D)r Fw(\()p Fr(A)p Fw(\))i(=)f Fq(D)r Fw(\()p Fr(A)p Fw(\).)83 2958 y(Therefore,)j(for)h(eac)n(h)f Fr(n)e Fq(2)f Fm(N)1004 2928 y Fp(\003)1075 2958 y Fw(the)28 b(op)r(erator)1567 2925 y Fy(1)p 1563 2939 42 4 v 1563 2987 a Fn(n)1629 2896 y Fi(P)1716 2916 y Fn(n)p Fp(\000)p Fy(1)1716 2983 y Fn(j)s Fy(=0)1860 2958 y Fr(U)1925 2928 y Fp(\000)p Fn(j)2010 2891 y Fi(\002)2045 2958 y Fr(A;)14 b(U)2208 2928 y Fn(j)2242 2891 y Fi(\003)2304 2958 y Fw(is)27 b(b)r(ounded,)e(and)i(the)g(op)r(erator)497 3235 y Fr(A)558 3247 y Fn(n)604 3235 y Fr(')c Fw(:=)808 3179 y(1)p 803 3216 53 4 v 803 3292 a Fr(n)879 3131 y Fn(n)p Fp(\000)p Fy(1)882 3156 y Fi(X)885 3335 y Fn(j)s Fy(=0)1019 3235 y Fr(U)1084 3201 y Fp(\000)p Fn(j)1168 3235 y Fr(AU)1294 3201 y Fn(j)1329 3235 y Fr(')f Fq(\021)1509 3179 y Fw(1)p 1504 3216 V 1504 3292 a Fr(n)1580 3131 y Fn(n)p Fp(\000)p Fy(1)1583 3156 y Fi(X)1586 3335 y Fn(j)s Fy(=0)1720 3235 y Fr(U)1785 3201 y Fp(\000)p Fn(j)1870 3168 y Fi(\002)1904 3235 y Fr(A;)14 b(U)2067 3201 y Fn(j)2101 3168 y Fi(\003)2136 3235 y Fr(')k Fw(+)g Fr(A';)180 b(')22 b Fq(2)i(D)r Fw(\()p Fr(A)2926 3247 y Fn(n)2973 3235 y Fw(\))f(:=)g Fq(D)r Fw(\()p Fr(A)p Fw(\))p Fr(;)83 3504 y Fw(is)k(self-adjoin)n(t.)e(F)-7 b(urthermore,)24 b(w)n(e)k(ha)n(v)n(e)e(the)i(follo)n(wing)c(result)j(on)f(the)i (regularit)n(y)c(of)k Fr(U)36 b Fw(with)27 b(resp)r(ect)g(to)h Fr(A)3630 3516 y Fn(n)3676 3504 y Fw(:)p Black 83 3678 a Ft(Lemma)j(4.1.)p Black 40 w Fx(T)-7 b(ake)32 b Fr(n)23 b Fq(2)h Fm(N)1036 3648 y Fp(\003)1111 3678 y Fx(and)31 b(let)g Fr(A)h Fx(and)f Fr(U)40 b Fx(b)-5 b(e)32 b(r)-5 b(esp)g(e)g(ctively)33 b(a)e(self-adjoint)g(and)g(a)h(unitary)f(op)-5 b(er)g(ator)83 3786 y(in)32 b(a)g(Hilb)-5 b(ert)32 b(sp)-5 b(ac)g(e)32 b Fq(H)q Fx(.)g(Then,)p Black 131 3960 a(\(a\))p Black 42 w(if)41 b Fr(U)31 b Fq(2)24 b Fr(C)617 3930 y Fy(1)654 3960 y Fw(\()p Fr(A)p Fw(\))p Fx(,)33 b(then)f Fr(U)f Fq(2)23 b Fr(C)1268 3930 y Fy(1)1306 3960 y Fw(\()p Fr(A)1399 3972 y Fn(n)1446 3960 y Fw(\))32 b Fx(with)f Fw([)p Fr(A)1787 3972 y Fn(n)1834 3960 y Fr(;)14 b(U)9 b Fw(])22 b(=)2083 3927 y Fy(1)p 2079 3941 42 4 v 2079 3989 a Fn(n)2144 3898 y Fi(P)2232 3918 y Fn(n)p Fp(\000)p Fy(1)2232 3985 y Fn(j)s Fy(=0)2376 3960 y Fr(U)2441 3930 y Fp(\000)p Fn(j)2526 3960 y Fw([)p Fr(A;)14 b(U)9 b Fw(])p Fr(U)2800 3930 y Fn(j)2833 3960 y Fx(,)p Black 135 4134 a(\(b\))p Black 43 w(if)41 b Fr(U)31 b Fq(2)24 b Fr(C)617 4104 y Fy(1+0)738 4134 y Fw(\()p Fr(A)p Fw(\))p Fx(,)33 b(then)f Fr(U)f Fq(2)23 b Fr(C)1352 4104 y Fy(1+0)1474 4134 y Fw(\()p Fr(A)1567 4146 y Fn(n)1614 4134 y Fw(\))p Fx(.)p Black 83 4309 a(Pr)-5 b(o)g(of.)p Black 42 w Fw(\(a\))28 b(Since)e Fr(U)774 4279 y Fn(k)837 4309 y Fq(2)d Fr(C)980 4279 y Fy(1)1018 4309 y Fw(\()p Fr(A)p Fw(\))29 b(for)e(eac)n(h)f Fr(k)e Fq(2)g Fm(Z)p Fw(,)e(one)27 b(has)f(for)h(an)n(y)g Fr(')c Fq(2)g(D)r Fw(\()p Fr(A)2638 4321 y Fn(n)2685 4309 y Fw(\))28 b(that)220 4503 y Fi(\012)259 4570 y Fr(A)320 4582 y Fn(n)367 4570 y Fr(';)14 b(U)9 b(')579 4503 y Fi(\013)635 4570 y Fq(\000)718 4503 y Fi(\012)757 4570 y Fr(';)14 b(U)9 b(A)975 4582 y Fn(n)1020 4570 y Fr(')1075 4503 y Fi(\013)1137 4570 y Fw(=)1240 4514 y(1)p 1234 4551 53 4 v 1234 4627 a Fr(n)1311 4467 y Fn(n)p Fp(\000)p Fy(1)1314 4492 y Fi(X)1316 4670 y Fn(j)s Fy(=0)1450 4503 y Fi(\012)1490 4570 y Fr(';)1582 4503 y Fi(\000)1619 4570 y Fr(U)1684 4536 y Fp(\000)p Fn(j)1769 4570 y Fr(AU)1895 4536 y Fn(j)s Fy(+1)2032 4570 y Fq(\000)18 b Fr(U)2180 4536 y Fy(1)p Fp(\000)p Fn(j)2298 4570 y Fr(AU)2424 4536 y Fn(j)2458 4503 y Fi(\001)2496 4570 y Fr(')2551 4503 y Fi(\013)2613 4570 y Fw(=)2716 4514 y(1)p 2710 4551 V 2710 4627 a Fr(n)2786 4467 y Fn(n)p Fp(\000)p Fy(1)2789 4492 y Fi(X)2792 4670 y Fn(j)s Fy(=0)2926 4503 y Fi(\012)2965 4570 y Fr(';)c(U)3122 4536 y Fp(\000)p Fn(j)3207 4570 y Fw([)p Fr(A;)g(U)9 b Fw(])p Fr(U)3481 4536 y Fn(j)3514 4570 y Fr(')3569 4503 y Fi(\013)3608 4570 y Fr(;)83 4843 y Fw(with)28 b([)p Fr(A;)14 b(U)9 b Fw(])22 b Fq(2)i Fj(B)s Fw(\()p Fq(H)q Fw(\).)29 b(This)d(implies)f(the)j(claim.)249 4951 y(\(b\))g(W)-7 b(e)28 b(kno)n(w)f(from)f(p)r(oin)n(t)g(\(a\))i (that)f Fr(U)k Fq(2)24 b Fr(C)1705 4921 y Fy(1)1742 4951 y Fw(\()p Fr(A)1835 4963 y Fn(n)1882 4951 y Fw(\))k(with)f([)p Fr(A)2216 4963 y Fn(n)2263 4951 y Fr(;)14 b(U)9 b Fw(])22 b(=)2512 4919 y Fy(1)p 2507 4933 42 4 v 2507 4980 a Fn(n)2573 4889 y Fi(P)2661 4910 y Fn(n)p Fp(\000)p Fy(1)2661 4976 y Fn(j)s Fy(=0)2805 4951 y Fr(U)2870 4921 y Fp(\000)p Fn(j)2955 4951 y Fw([)p Fr(A;)14 b(U)9 b Fw(])p Fr(U)3229 4921 y Fn(j)3262 4951 y Fw(.)28 b(Therefore,)d(it)83 5059 y(is)i(su\016cien)n(t)f(to)i(sho)n(w)e(for)h(eac)n(h)g Fr(j)h Fq(2)23 b(f)p Fw(1)p Fr(;)14 b(:)g(:)g(:)f(;)h(n)k Fq(\000)g Fw(1)p Fq(g)27 b Fw(that)1291 5188 y Fi(Z)1374 5209 y Fy(1)1337 5377 y(0)1435 5245 y Fw(d)p Fr(t)p 1435 5282 77 4 v 1458 5358 a(t)1531 5231 y Fi(\015)1531 5281 y(\015)1591 5299 y Fw(e)1627 5267 y Fp(\000)p Fn(itA)1774 5275 y Fg(n)1833 5301 y Fr(B)1896 5313 y Fn(j)1944 5299 y Fw(e)1981 5267 y Fn(itA)2077 5275 y Fg(n)2136 5301 y Fq(\000)p Fr(B)2264 5313 y Fn(j)2297 5231 y Fi(\015)2297 5281 y(\015)2367 5301 y Fr(<)22 b Fq(1)p Fr(;)p Black 1884 5670 a Fw(13)p Black eop end %%Page: 14 14 TeXDict begin 14 13 bop Black Black 83 307 a Fw(with)28 b Fr(B)337 319 y Fn(j)394 307 y Fw(:=)22 b Fr(U)569 277 y Fp(\000)p Fn(j)654 307 y Fw([)p Fr(A;)14 b(U)9 b Fw(])p Fr(U)928 277 y Fn(j)962 307 y Fw(.)27 b(But,)h(for)f(eac)n(h)g Fr(t)c Fq(2)g Fm(R)35 b Fw(and)26 b(eac)n(h)h Fr(')22 b Fq(2)i(D)r Fw(\()p Fr(A)p Fw(\))29 b(w)n(e)e(ha)n(v)n(e)227 570 y(e)264 538 y Fn(itA)360 546 y Fg(n)418 572 y Fr(')18 b Fq(\000)574 570 y Fw(e)611 538 y Fn(itA)726 572 y Fr(')23 b Fw(=)891 459 y Fi(Z)974 479 y Fn(t)937 647 y Fy(0)1017 572 y Fw(d)p Fr(s)1146 516 y Fw(d)p 1127 553 87 4 v 1127 629 a(d)p Fr(s)1223 505 y Fi(\000)1274 570 y Fw(e)1311 538 y Fn(isA)1414 546 y Fg(n)1473 570 y Fw(e)1509 538 y Fp(\000)p 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