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

Let T={T(t)}t∈R be a C0-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 (cn)n∈Z and (tn)n∈Z such thatAx=∑n∈ZcnT(tn)x, where∑n∈Z|cn|=rA(x); rA(x) is the local spectral radius of A at x. As an application, some inequalities of Bernstein type in Lp-spaces are given.