Perturbation analysis for the generalized inverses with prescribed idempotents in Banach algebras

TL;DR

This paper investigates how generalized inverses with prescribed idempotents in Banach algebras behave under perturbations, providing formulas and error estimates for their stability in the presence of small changes.

Contribution

It introduces new perturbation formulas and stability analysis for various types of generalized inverses with prescribed idempotents in Banach algebras.

Findings

Derived explicit formulas for perturbed generalized inverses

Established error bounds for small perturbations

Analyzed stability conditions for these inverses

Abstract

In this paper, we first study the perturbations and expressions for the generalized inverses $a^{(2)}_{p,q}$, $a^{(1, 2)}_{p,q}$, $a^{(2, l)}_{p,q}$ and $a^{(l)}_{p,q}$ with prescribed idempotents $p$ and $q$. Then, we investigate the general perturbation analysis and error estimate for some of these generalized inverses when $p,\,q$ and $a$ also have some small perturbations.