B-Convexity, the Analytic Radon–Nikodym Property, and Individual Stability ofC0-Semigroups

Abstract

LetT={T(t)}t≥0be aC0-semigroup on a Banach spaceX, with generatorAand growth bound ω. Assume thatx0∈Xis such that the local resolvent λ↦R(λ,A)x0admits a bounded holomorphic extension to the right half-plane {Reλ>0}. We prove the following results:•(i) Ifhas Fourier type∈(1,2], then lim‖()(λ−)‖=0 for all β>1/and λ>ω.•(ii) Ifhas the analytic RNP, then lim‖()(λ−)‖=0 for all β>1 and λ>ω.•(iii) Ifis arbitrary, then weak-lim()(λ−)=0 for all β>1 and λ>ω.As an application we prove a Tauberian theorem for the Laplace transform of functions with values in aB-convex Banach space.