Cline's formula for G-Drazin inverse

TL;DR

This paper extends Clines formula for the G-Drazin inverse to a broader class of operators satisfying specific algebraic conditions, with applications to Banach space operators.

Contribution

It generalizes Clines formula for the G-Drazin inverse under new algebraic conditions, expanding its applicability.

Findings

Extended Clines formula for G-Drazin inverse under new conditions

Applications demonstrated on Banach space operators

Broadened understanding of generalized inverses in operator theory

Abstract

Clines formula for the well known generalized inverses such as Drazin inverse, generalized Drazin inverse is extended to the case when $a(ba)^2=abaca=acaba=(ac)^2a$ . Applications are given to some interesting Banach space operators.