Generalized right group inverse in Banach *-algebras

TL;DR

This paper introduces a new type of inverse called the generalized right group inverse in Banach *-algebras, extending existing concepts by combining right group inverses with quasinilpotency, and explores its properties and relationships.

Contribution

It defines and characterizes the generalized right group inverse in Banach *-algebras, extending the theory of generalized (weak) group inverses and their relations.

Findings

Characterization of the generalized right group inverse

Representation formulas for the inverse

Relationship with the generalized right EP-inverse

Abstract

In this paper, we introduce the concept of the generalized right group inverse within the context of a *-Banach algebra. This represents a natural extension of the generalized (weak) group inverse. Notably, this generalized inverse is characterized by integrating the right group inverse with the concept of quasinilpotency. We provide various characterizations and representations of the generalized right group inverse. Furthermore, we explore the relationship between the generalized right group inverse and the generalized right EP-inverse. The properties of the generalized (weak) group inverse in a Banach *-algebra are also extended to a more general framework.