One-sided Drazin inverses in Banach algebras and perturbations of B-Fredholm spectra

TL;DR

This paper introduces one-sided Drazin inverses in Banach algebras, characterizes their spectra, and extends classical Fredholm theory to include perturbation results for B-Fredholm spectra.

Contribution

It defines and characterizes left and right (generalized) Drazin inverses in Banach algebras, extending Fredholm theory with new spectral perturbation results.

Findings

Characterizations of one-sided (generalized) Drazin invertible operators

Spectral descriptions of B-Fredholm operators

Perturbation results extending classical Fredholm theory

Abstract

The famous Drazin inverse and generalized Drazin inverse were introduced by Drazin in 1958 and Koliha in 1996, respectively. In the present paper, the author introduces the concepts of left and right (generalized) Drazin inverses, which are the one-sided versions of classical (generalized) Drazin inverses, in Banach algebras. Several characterizations of one-sided (generalized) Drazin invertible operators on Banach spaces are given. By utilizing the one-sided Drazin invertible spectra, the characterizations of B-Fredholm spectra for Banach space operators are obtained. These perturbational results can be regarded as generalizations of classical Fredholm theory in Banach spaces.