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

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.