Co-EP Banach Algebra Elements

TL;DR

This paper characterizes the invertibility of the commutator of an element and its Moore-Penrose inverse in Banach algebras, studies a special subset, perturbations, and extends results to Banach space operators.

Contribution

It provides new characterizations of invertibility for a specific commutator in Banach algebras and explores perturbations and operator cases.

Findings

Characterization of invertibility of $aa^ - a^ a$

Analysis of a special subset of elements with this property

Results on perturbations and operator extensions

Abstract

In this work, given a unital Banach algebra $\A$ and $a\in \A$ such that $a$ has a Moore-Penrose inverse $a^\dagger$, it will be characterized when $aa^\dagger-a^\dagger a$ is invertible. A particular subset of this class of objects will also be studied. In addition, perturbations of this class of elements will be studied. Finally, the Banach space operator case will be also considered.