Left and right generalized Drazin invertible operators and Local spectral theory

TL;DR

This paper characterizes left and right generalized Drazin invertible operators in Banach spaces using the SVEP, explores their stability under perturbations, and examines how local spectral properties transfer to their inverses.

Contribution

It provides new characterizations of generalized Drazin invertibility via SVEP and analyzes the stability of these operators and spectral properties under perturbations.

Findings

Characterizations of generalized Drazin invertibility using SVEP.

Stability of these operators under finite rank perturbations.

Transfer of local spectral properties to generalized Drazin inverses.

Abstract

In this paper, we give some characterizations of the left and right generalized Drazin invertible bounded operators in Banach spaces by means of the single-valued extension property (SVEP). In particular, we show that a bounded operator is left (resp. right) generalized Drazin invertible if and only if admits a generalized Kato decomposition and has the SVEP at 0 (resp. it admits a generalized Kato decomposition and its adjoint has the SVEP at 0. In addition, we prove that both of the left and the right generalized Drazin operators are invariant under additive commuting finite rank perturbations. Furthermore, we investigate the transmission of some local spectral properties from a bounded linear operator, as the SVEP, Dunford property $(C)$, and property $(\beta)$, to its generalized Drazin inverse.