B-Convexity, the Analytic Radon–Nikodym Property, and Individual Stability of C0-Semigroups

Let T D ”Tt‘•t 0 be a C0-semigroup on a Banach space X, with generator A and growth bound !. Assume that x0 2 X is such that the local resolvent 7! R ;A‘x0 admits a bounded holomorphic extension to the right half-plane ”Re >0•. We prove the following results: (i) If X has Fourier type p 2 1; 2“, then limt!1 Tt‘ 0 − A‘− x0 D 0 for all > 1=p and 0 > !. (ii) If X has the analytic RNP, then limt!1 Tt‘ 0 − A‘− x0 D 0 for all > 1 and 0 > !. (iii) If X is arbitrary, then weak-limt!1 Tt‘ 0 − A‘− x0 D 0 for all > 1 and 0 > !. As an application we prove a Tauberian theorem for the Laplace transform of functions with values in a B-convex Banach space.