Remark on common properties of the products $ac$ and $ba$

TL;DR

This paper explores common algebraic properties of products $ac$ and $ba$ under specific conditions, extending classical lemmas to generalized invertibility in rings and operators on Banach spaces.

Contribution

It generalizes Jacobson's lemma and Cline's formula for n-strong Drazin invertibility and Fredholm operators, broadening their applicability.

Findings

Generalized Jacobson's lemma applies to left and right Fredholm operators.

Cline's formula extends to generalized n-strong Drazin invertible rings.

Identifies common properties of products $ac$ and $ba$ under specific algebraic conditions.

Abstract

In this paper, we discuss the common properties for the products $ac$ and $ba$ in various categories under the condition $a(ba)^{2}=abaca=acaba=(ac)^{2}a$. We prove that generalized Jacobson's lemma and Cline's formula are suitable for generalized n-strong Drazin invertible in rings, and generalized Jacobson's lemma is suitable for left and right Fredholm operator on Banach spaces.