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

TL;DR

This paper extends spectral inclusion results for $C_0$-semigroups to Drazin invertible and Quasi-Fredholm spectra, providing conditions linking these properties of the generator and the semigroup.

Contribution

It establishes spectral inclusion for these spectra in $C_0$-semigroups and identifies conditions under which the properties of the generator are equivalent.

Findings

Spectral inclusion holds for Drazin invertible and Quasi-Fredholm spectra.

Conditions are provided for equivalence of quasi-Fredholm, Drazin invertible, and B-Fredholm properties of the generator.

The results generalize spectral theory for $C_0$-semigroups and their generators.

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.