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

Let A be a positive self-adjoint operator and let B be anm-accretive operator which is A-small with a relative bound less than one. LetH=A+B, thenH is well-defined on dom(H)=dom(A) andm-accretive. If B is a strictlym-accretive operator obeying…((1)) then for the Trotter product formula we prove that …(2))(and similar forH *) asn-(infinity), uniformly in t >0. We also show that: (a) the A-smallness of B guarantees the condition (1) for (alpha) (element of) (0,1/2), i.e. the estimate (2) holds for (alpha)(element of) (0,1/2); (b) if B is strictlym-sectorial, then there are sufficient conditions ensuring the relation (1) for (alpha)=1/2, that implies (2); (c) if B is A-small,m-sectorial and such that dom(A 1/2) is a subset of the formdomain of B, then again (2) is valid for (alpha)=1/2.