Perturbations of differentiable semigroups

Abstract

If (A, D(A)) generates a C 0-semigroup T on a Banach space X and $$B \in {\mathcal{L}}(X)$$ then (A + B, D(A)) is also the generator of a C 0-semigroup, S B . There are easy examples to show that if T is eventually differentiable then S B need not be eventually differentiable. In 1995 an example was constructed to show that if T is immediately differentiable then S B need not be immediately differentiable. In this paper we establish necessary and sufficient conditions on the generator (A, D(A)) of T which ensure that eventual or immediate differentiability of T is inherited by S B for all $$B \in {\mathcal{L}}(X)$$ . We are therefore able to give a characterization of the immediately and eventually differentiable C 0-semigroups for which differentiability is a stable property under bounded perturbations of the generator. We also prove a characterization of the C 0-semigroups for which the norm of the resolvent of the generator decays on vertical lines and a new characterization of the Crandall-Pazy class of semigroups.