Pseudo semi B-Fredholm and Generalized Drazin invertible operators Through Localized SVEP

TL;DR

This paper explores the properties of pseudo semi B-Fredholm and generalized Drazin invertible operators in Banach spaces, establishing spectral equalities, invariance under perturbations, and applications to operator matrices.

Contribution

It introduces new spectral characterizations and invariance results for pseudo semi B-Fredholm and generalized Drazin invertible operators, including their behavior under perturbations and applications to operator matrices.

Findings

Equality between left generalized Drazin spectrum and pseudo upper semi B-Fredholm spectrum up to $S(T)$

Invariance of generalized Drazin operators under additive commuting power finite rank perturbations

Conditions for spectra of upper triangular operator matrices to be unions of diagonal entries' spectra

Abstract

In this paper, we define and study the pseudo upper and lower semi B-Fredholm of bounded operators in a Banach space. In particular, we prove equality up to $S(T)$ between the left generalized Drazin spectrum and the pseudo upper semi B-Fredholm spectrum, $S(T )$ is the set where $T$ fails to have the SVEP. Also, we prove that both of the left and the right generalized Drazin operators are invariant under additive commuting power finite rank perturbations and some perturbations for the pseudo upper and lower semi B-Fredholm operators are given. As applications, we investigate some classes of operators as the supercyclic and multiplier operators. Furthermore, we investigate the left and the right generalized Drazin invertibility of upper triangular operator matrices by giving sufficient conditions which assure that the left and the right generalized Drazin spectrum or the pseudo upper and lower semi B-Fredholm of an upper triangular operator matrices is the union of its diagonal entries spectra.