 0194 Cachia V., Neidhardt H., Zagrebnov V.
 Comments on the Trotter Product Formula ErrorBound Estimates
for NonselfAdjoint Semigroups.
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Mar 7, 01

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Abstract. Let $A$ be a positive selfadjoint operator and let $B$ be an
$m$accretive operator which is $A$small with a relative bound
less than one. Let $H = A + B$, then $H$ is welldefined on ${\rm
dom}(H) = {\rm dom}(A)$ and $m$accretive. If $B$ is a strictly
$m$accretive operator obeying
\begin{equation}\label{aa}
{\rm dom}((H^{\ast })^{\alpha })\subseteq {\rm dom}(A^{\alpha })\cap
{\rm dom}((B^{\ast })^{\alpha })\neq \{0\} \quad
\mbox{for some}\quad \alpha \in (0,1],
\end{equation}
then for the Trotter product formula we prove that
\begin{equation} \label{bb}
\left\ \left( e^{tB/n}e^{tA/n}\right) ^{n}e^{tH}\right\ \wedge
\left\ \left( e^{tA/n}e^{tB/n}\right) ^{n}e^{tH}\right\ =
O(\ln n/n^\alpha)
\end{equation}
(and similar for $H^{\ast }$) as $n \to \infty$, uniformly in
$t\geq0$. We also show that:\\ (a) the $A$smallness of $B$
guarantees the condition (\ref{aa}) for $\alpha \in (0,1/2)$,
i.e. the estimate (\ref{bb}) holds for $\alpha \in (0,1/2)$;\\
(b) if $B$ is strictly $m$sectorial, then there are sufficient
conditions ensuring the relation (\ref{aa}) for $\alpha = 1/2$,
that implies (\ref{bb});\\
(c) if $B$ is $A$small, $m$sectorial and such that ${\rm
dom}(A^{1/2})$ is a subset of the formdomain of $B$, then again
(\ref{bb}) is valid for $\alpha = 1/2$.
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