Spectral inclusion for $C_0$-Semigroups Drazin invertible and   Quasi-Fredholm operators
  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. 
A. Tajmouati, M. Amouch, and M. R. F. Z. Alhomidi
math.SP