Ichinose T., Neidhardt H., Zagrebnov V.A.
Trotter-Kato product formula and fractional powers of
self-adjoint generators
(207K, Postscript)
ABSTRACT. Let $A$ and $B$ be non-negative self-adjoint 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 non-symmetric Trotter-Kato 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\ga-1)}), \\[2mm]
\left\|\left(e^{-tA/n}e^{-tB/n}\right)^n - e^{-tH}\right\| =
O(n^{-(2\ga-1)})
\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 real-valued Borel
measurable functions $f(\cdot)$ defined on the non-negative 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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