Local Spectral Properties of Generators of C0-Groups and Bernstein Type Inequalities

Abstract

Let T = {T(t)}(t epsilon R) be a C-0-group on a complex Banach space X dominated by a weight function omega(t) = (1 + vertical bar t vertical bar)(alpha) (0 <= alpha < 1) and let A be its generator with domain D(A). Among other things, it is shown that if the operator A has compact local spectrum at x is an element of X, then x is an element of D(A) and there exist double sequences of real numbers and (t(n))(n epsilon Z) Such that Ax = Sigma(n is an element of Z) CnT(t(n))x. where Sigma(n is an element of Z) vertical bar C-n vertical bar = r(A)(x); r(A)(X) is the local spectral radius of A at x. As an application, some inequalities of Bernstein type in L-P-spaces are given. (C) 2009 Elsevier Inc. All rights reserved.