Extensions of Jacobson's lemma for generalized inverses in a ring

TL;DR

This paper extends Jacobson's lemma to generalized inverses in rings, establishing equivalences between the invertibility of certain elements and exploring spectral properties in Banach algebras.

Contribution

It introduces new conditions under which the generalized Drazin inverse exists for elements related by specific algebraic relations, extending Jacobson's lemma.

Findings

Equivalence of generalized Drazin invertibility for 1 - ac and 1 - ba

New spectral properties for ac and ba in Banach algebras

Extension of Jacobson's lemma to B-Fredholm and generalized Fredholm elements

Abstract

Let $R$ be an associative ring with unit $1$, and $a, b, c\in R$ satisfy $a(ba)^{2}=abaca=acaba=(ac)^{2}a$, this paper proves that $1-ac$ has generalized Drazin inverse (Drazin inverse, pseudo Drazin inverse, respectively) if and only if $1-ba$ has generalized Drazin inverse (Drazin inverse, pseudo Drazin inverse, respectively). In particular, we obtain new common spectral properties for $ac$ and $ba$ in Banach algebras. As applications, new extension of Jacobson's lemma for B-Fredholm elements and generalized Fredholm elements in rings is established.