Perturbations and expressions of the Moore--Penrose metric generalized inverses and applications to the stability of some operator equations

TL;DR

This paper studies how the Moore--Penrose metric generalized inverse of bounded linear operators on Banach spaces behaves under perturbations, providing conditions for simplified expressions and analyzing the stability of operator equations.

Contribution

It offers new geometric conditions for the explicit expression of the Moore--Penrose inverse of perturbed operators and applies these results to stability analysis of operator equations.

Findings

Derived equivalent conditions for the Moore--Penrose inverse expression.

Established stability results for operator equations under perturbations.

Provided geometric assumptions that simplify inverse calculations.

Abstract

In this paper, the problems of perturbation and expression for the Moore--Penrose metric generalized inverses of bounded linear operators on Banach spaces are further studied. By means of certain geometric assumptions of Banach spaces, we first give some equivalent conditions for the Moore--Penrose metric generalized inverse of perturbed operator to have the simplest expression $T^M(I+ \delta TT^M)^{-1}$. Then, as an application our results, we investigate the stability of some operator equations in Banach spaces under different type perturbations.