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 {Re λ > 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.