The $m$-generalized right group inverses in Banach algebras

TL;DR

This paper introduces the m-generalized right group inverse in Banach algebras, extending existing concepts and providing new characterizations and representations that deepen understanding of generalized inverses.

Contribution

It defines the m-generalized right group inverse, characterizes it via decompositions and properties, and links it to the generalized right core inverse, advancing the theory of generalized inverses.

Findings

Introduction of the m-generalized right group inverse

Characterization through m-generalized right group decomposition

Representation using the generalized right core inverse

Abstract

In this paper, we introduce the concept of the m-generalized right group inverse. This serves as a natural extension of both the m-weak group inverse and the generalized group inverse. We characterize this new generalized inverse using the m-generalized right group decomposition and a polar-like property. Additionally, we present the representation of the m-generalized right group inverse using the generalized right core inverse, leading to new insights and properties for both the m-weak group inverse and the generalized group inverse.