Jacobson's Lemma for the generalized n-strongly Drazin inverse

TL;DR

This paper extends Jacobson's Lemma to the generalized n-strongly Drazin inverse in rings and Banach algebras, establishing an equivalence condition for the invertibility of 1-ab and 1-ba.

Contribution

It introduces the concept of generalized n-strongly Drazin inverse and proves a key symmetry property in rings and Banach algebras.

Findings

Proves the equivalence of invertibility of 1-ab and 1-ba for the generalized n-strongly Drazin inverse.

Extends the result to Banach algebras.

Provides a new framework for analyzing generalized inverses in algebraic structures.

Abstract

Let $n\in {\Bbb N}$. An element $a\in R$ has generalized n-strongly Drazin inverse if there exists $x\in R$ such that $xax=x, x\in comm^2(a), a^n-ax\in R^{qnil}.$ For any $a,b\in R$, we prove that $1-ab$ has generalized n-strongly Drazin inverse if and only if $1-ba$ has generalized n-strongly Drazin inverse. Extensions in Banach algebra are also obtained.