Comments on the Trotter product formula error-bound estimates for nonself-adjoint semigroups

Abstract

LetA be a positive self-adjoint operator and letB be anm-accretive operator which isA-small with a relative bound less than one. LetH=A+B, thenH is well-defined on dom(H)=dom(A) andm-accretive. IfB is a strictlym-accretive operator obeying $$dom((H^ * )^\alpha ) \subseteq dom(A^\alpha ) \cap dom((B^ * )^\alpha ) \ne \{ 0\} for some\alpha \in (0,1],$$ (1) then for the Trotter product formula we prove that $$\left\| {(e^{{{ - tB} \mathord{\left/ {\vphantom {{ - tB} n}} \right. \kern-\nulldelimiterspace} n}} e^{{{ - tA} \mathord{\left/ {\vphantom {{ - tA} n}} \right. \kern-\nulldelimiterspace} n}} )^n - e^{tH} } \right\| \wedge \left\| {(e^{{{ - tA} \mathord{\left/ {\vphantom {{ - tA} n}} \right. \kern-\nulldelimiterspace} n}} e^{{{ - tB} \mathord{\left/ {\vphantom {{ - tB} n}} \right. \kern-\nulldelimiterspace} n}} )^n - e^{tH} } \right\| = O(In{n \mathord{\left/ {\vphantom {n {n^\alpha }}} \right. \kern-\nulldelimiterspace} {n^\alpha }})$$ (2) (and similar forH*) asn→∞, uniformly int≥0. We also show that: (a) theA-smallness ofB guarantees the condition (1) for α∈(0,1/2), i.e. the estimate (2) holds for α∈(0,1/2); (b) ifB is strictlym-sectorial, then there are sufficient conditions ensuring the relation (1) for α=1/2, that implies (2); (c) ifB isA-small,m-sectorial and such that dom(A1/2) is a subset of the formdomain ofB, then again (2) is valid for α=1/2.