The characterizations and representations for the generalized inverses with prescribed idempotents in Banach algebras

TL;DR

This paper explores new characterizations and explicit formulas for generalized inverses in Banach algebras with prescribed idempotents, extending known results from matrices and operators.

Contribution

It introduces novel characterizations and representations for generalized inverses with prescribed idempotents in Banach algebras, broadening existing mathematical frameworks.

Findings

Derived explicit formulas for generalized inverses with prescribed idempotents.

Extended known results from matrices and operators to Banach algebras.

Provided new characterizations for various types of generalized inverses.

Abstract

In this paper, we investigate the various different generalized inverses in a Banach algebra with respect to prescribed two idempotents $p$ and $q$. Some new characterizations and explicit representations for these generalized inverses, such as $a^{(2)}_{p,q}$, $a^{(1,2)}_{p,q}$ and $a^{(2,l)}_{p,q}$ will be presented. The obtained results extend and generalize some well--known results for matrices or operators.