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

2020/06/11 by Chen, Huanyin, Abdolyousefi, Marjan Sheibani
#FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2006.06720

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∈ R satisfying (ac)² = (db)(ac), (db)² = (ac)(db);
b(ac)a = b(db)a, c(ac)d = c(db)d. Then ac∈ R^d if and only if bd∈ 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.

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• Expressions for the g-Drazin inverse in a Banach algebra