Spectral inclusion for $C_0$-Semigroups Drazin invertible and Quasi-Fredholm operators

Abstract: Let $(T(t)){t\geq0}$ be a $C_0$ semigroups and $A$ be its infinitesimal generator. In this work, we prove that the spectral inclusion for $(T(t)){t\geq0}$ remains true for the Drazin invertible and Quasi-Fredholm spectra. Also, we will give conditions under which facts $A$ is quasi-Fredholm, $A$ is Drazin invertible and $A$ is B-Fredholm are equivalent.