Generalized cline’s formula for g-Drazin inverse in a ring

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