Some spectral inclusion for strongly continuous semigroups operators

Abstract

Let (T(t))_t≥0 be a C₀-semigroup on a Banach space X.  In this paper, we show that if there exists t₀ > 0 such that T(t₀) is a pseudo B-Fredholm operator, then T(t) is pseudo B-Fredholm for all t ≥ 0, which is equivalent that T(t) is generalized Drazin invertible for all t ≥ 0.  Also we prove that the spectral inclusion of strongly continuous semigroup hold for pseudo Fredholm, generalized Drazin and pseudo B-Fredholm spectra.