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 ∈ R satisfyingThen 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