The characterizations of the stable perturbation of a closed operator by a linear operator in Banach spaces

TL;DR

This paper studies the stability of closed operators under perturbations in Banach spaces, providing new characterizations of when such perturbations preserve invertibility and extending recent results in the field.

Contribution

It introduces new conditions for the invertibility of perturbed closed operators in Banach spaces, generalizing previous findings by Huang.

Findings

Characterization of invertibility of I_Y + δTT^+

Conditions for stable perturbations of closed operators

Extension of Huang's recent results

Abstract

In this paper, we investigate the invertibility of $I_Y+\delta TT^+$ when $T$ is a closed operator from $X$ to $Y$ with a generalized inverse $T^+$ and $\delta T$ is a linear operator whose domain contains $D(T)$ and range is contained in $D(T^+)$. The characterizations of the stable perturbation $T+\delta T$ of $T$ by $\delta T$ in Banach spaces are obtained. The results extend the recent main results of Huang's in Linear Algebra and its Applications.