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.
Similar content being viewed by others
Author information
Authors and Affiliations
Corresponding author
Additional information
We are grateful to Charles Batty and Tom Ransford for helpful discussions and to the referee for their constructive comments.
Rights and permissions
About this article
Cite this article
Iley, P.S. Perturbations of differentiable semigroups. J. evol. equ. 7, 765–781 (2007). https://doi.org/10.1007/s00028-007-0349-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-007-0349-0