Local spectral properties of generators of C0-groups and Bernstein type inequalities

Abstract

Let T = {T (t)} t∈R be a C 0 -group on a complex Banach space X dominated by a weight function ω(t) = (1 + |t|) α (0 α < 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 ∈ X, then x ∈ D( A) and there exist double sequences of real numbers (c n ) n∈Z and (t n ) n∈Z such thatis the local spectral radius of A at x. As an application, some inequalities of Bernstein type in L p -spaces are given.