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 ? R satisfying (ac)2 = (db)(ac), (db)2 = (ac)(db), b(ac)a = b(db)a, c(ac)d = c(db)d. Then ac ? Rd if and only if bd ? Rd. 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., 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 is obtained.