Generalized Cline's formula for G-Drazin inverse in a ring

TL;DR

This paper extends Cline's formula to the generalized Drazin inverse in rings, providing new conditions and formulas, and explores their implications for spectral properties of operators in Banach spaces.

Contribution

It introduces a generalized Cline's formula for the g-Drazin inverse in rings, expanding previous results and applying them to spectral theory in Banach spaces.

Findings

Generalized Cline's formula for g-Drazin inverse in rings.

Equivalent conditions for invertibility of products in rings.

New spectral properties for bounded linear operators.

Abstract

In this paper, we give a generalized Cline's formula for the generalized Drazin inverse. Let $R$ be a ring, and let $a,b,c,d\in R$ satisfying $$\begin{array}{c} (ac)^2 = (db)(ac), (db)^2 = (ac)(db);\\ b(ac)a = b(db)a, c(ac)d = c(db)d.\end{array}$$ Then $ac\in R^d$ if and only if $bd\in R^d$. In this case, $(bd)^d = b((ac)^d)^2 d$. We also present generalized Cline's formulas for Drazin and group inverses. Some weaker conditions in a Banach algebra are also investigated. These extend the main results of Cline's formula on g-Drazin inverse of Liao, Chen and Cui (Bull. Malays. Math. Soc., 37 (2014), 37-42), Lian and Zeng (Turk. J. Math., 40 (2016), 161-165) and Miller and Zguitti (Rend. Circ. Mat. Palermo, II. Ser., 67 (2018), 105-114). As an application, new common spectral property of bounded linear operators over Banach spaces are obtained.