Generalized Jacobson's lemma in a Banach algebra

TL;DR

This paper extends Jacobson's lemma within Banach algebras by establishing conditions under which elements have g-Drazin inverses, providing explicit formulas and generalizing previous results.

Contribution

It generalizes the Jacobson's lemma for g-Drazin inverses in Banach algebras, offering new equivalences and explicit inverse formulas.

Findings

1 - ba is g-Drazin invertible iff 1 - ac is g-Drazin invertible

Explicit formula for (1 - ac)^d in terms of (1 - ba)^d and other elements

Extends previous work by Corach on g-Drazin inverses

Abstract

Let A be a Banach algebra, and let a; b; c 2 A satisfying a(ba)^2 = abaca = acaba = (ac)^2a: We prove that 1 - ba\in A^d if and only if 1 - ac \in A^d. In this case, (1-ac)^d =1-a(1-ba)^{\pi}(1-\alpha(1+ba))^{-1}bac (1+ac)+a((1-ba)^d)bac. This extends the main result on g-Drazin inverse of Corach (Comm. Algebra, 41(2013), 520{531).