m-generalized group inverse in Banach algebras

Abstract

In this paper, we introduce the notion of the $m$-generalized group inverse in a *-Banach algebra.This is a natural generalization of $m$-weak group inverse and generalized group inverse. We present elementary properties of this new generalized inverse and characterize it by the $m$-generalized group decomposition and the polar-like property. Furthermore, the relations between $m$-generalized group inverse and generalized group inverse are investigated. New properties of $m$-weak group inverse and generalized group inverse are thereby obtained.