B-convexity, the analytic radon-nikodym property, and individual stabilityof C-0-semigroups

Let T = {T(t)}(t greater than or equal to 0) be a C-0-semigroup on a Banach space X, with generator A and growth bound omega. Assume that x(0) is an e lement of X is such that the local resolvent lambda bar right arrow R(lambd a, A)x(0) admits a bounded holomorphic extension to the right half-plane {R e lambda > 0}. We prove the following results: (i) If X has Fourier type p is an element of (1, 2], then lim(t-->infinity) parallel to T(t)(lambda(0) - A)(-beta)x(0)parallel to = 0 for all beta > 1/ p and lambda(0) > omega. (ii) If X has the analytic RNP, then lim(t-->infinity)parallel to T(t)(lamb da(0) - A)(-beta)x(0)parallel to = 0 for all beta > 1 and lambda(0) > omega . (iii) If X is arbitrary, then weak-lim(t-->infinity) T(t)(lambda(0) - A)(-b eta)x(0) = 0 for all beta > 1 and lambda(0) > omega. As an application we prove a Tauberian theorem for the Laplace transform of functions with values in a B-convex Banach space. (C) 1999 Academic Press.