Generalized B-Fredholm Banach algebra elements

TL;DR

This paper characterizes and studies properties of generalized B-Fredholm elements in Banach algebras relative to a homomorphism, including their perturbation behavior, extending the theory of Fredholm elements.

Contribution

It introduces a comprehensive characterization of generalized B-Fredholm elements in Banach algebras and analyzes their main properties and perturbation stability.

Findings

Characterization of B-Fredholm and generalized B-Fredholm elements

Analysis of their algebraic and spectral properties

Perturbation stability results for these elements

Abstract

Given a (not necessarily continuous) homomorphism between Banach algebras $\T\colon\A\to\B$, an element $a\in\A$ will be said to be B-Fredholm (respectively generalized B-Fredholm) relative to $\T$, if $\T(a)\in \B$ is Drazin invertible (respectively Koliha-Drazin invertible). In this article, the aforementioned elements will be characterized and their main properties will be studied. In addition, perturbation properties will be also considered.