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

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 that A x = ∑ n ∈ Z c n T ( t n ) x , where ∑ n ∈ Z | c n | = 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.