01-94 Cachia V., Neidhardt H., Zagrebnov V.

Comments on the Trotter Product Formula Error-Bound Estimates for Nonself-Adjoint Semigroups. (71K, LATeX) Mar 7, 01

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Abstract. Let $A$ be a positive self-adjoint 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 well-defined 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 form-domain of $B$, then again (\ref{bb}) is valid for $\alpha = 1/2$.

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