 02411 Ichinose T., Neidhardt H., Zagrebnov V.A.
 TrotterKato product formula and fractional powers of
selfadjoint generators
(207K, Postscript)
Oct 2, 02

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Abstract. Let $A$ and $B$ be nonnegative selfadjoint operators in a
Hilbert space such that their densely defined form sum
$H = A \stackrel{\cdot}{+} B$ obeys
$\dom(H^\ga) \subseteq \dom(A^\ga) \cap \dom(B^\ga)$
for some $\ga \in (1/2,1)$. It is proved that if, in addition,
$A$ and $B$ satisfy $\dom(A^{1/2}) \subseteq \dom(B^{1/2})$, then
the symmetric and nonsymmetric TrotterKato product
formula converges in the operator norm:
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\bed
\ba{c}
\left\\left(e^{tB/2n}e^{tA/n}e^{tB/2n}\right)^n  e^{tH}\right\
= O(n^{(2\ga1)}), \\[2mm]
\left\\left(e^{tA/n}e^{tB/n}\right)^n  e^{tH}\right\ =
O(n^{(2\ga1)})
\ea
\eed
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uniformly in $t \in [0,T]$, $0 < T < \infty$, as $n \to \infty$, both with
the same optimal error bound.
The same is valid if
one replaces the exponential function
in the product by functions of the Kato class, that is, by realvalued Borel
measurable functions $f(\cdot)$ defined on the nonnegative real axis obeying
$0 \le f(x) \le 1$, $f(0) = 1$ and $f'(+0) = 1$, with some additional
smoothness property at zero. The present result
improves previous ones relaxing the smallness of $B^\ga$ with respect to
$A^\ga$ to the milder assumption
$\dom(A^{1/2}) \subseteq \dom(B^{1/2})$ and extending essentially
the admissible class of Kato functions.
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