generalized Cline's formula for g-Drazin inverse in a ring

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,
$(bd)^{d}=b((ac)^{d})^2d.$ 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., {\bf 37}(2014), 37-42), Lian and Zeng (Turk. J. Math., {\bf 40}(2016), 161-165) and Miller and Zguitti (Rend. Circ. Mat. Palermo, II. Ser., {\bf 67}(2018), 105-114). As an application, new common spectral property of bounded linear operators over Banach spaces are obtained.