B-convexity, the analytic radon-nikodym property, and individual stability of C0-semigroups

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:231

Article

Huang, SZ ; van Neerven, JMAM

关键词: C-0-semigroup;    individual stability;    resolvent estimates;    B-convex;    Fourier type;    analytic Radon-Nikodym property;    Tauberian theorems;   

DOI  :  10.1006/jmaa.1998.6211

来源: Elsevier

【 摘 要 】

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 element of X is such that the local resolvent lambda bar right arrow R(lambda, A)x(0) admits a bounded holomorphic extension to the right half-plane {Re 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)(lambda(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)(-beta)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.

【 授权许可】

Free