Perturbation of closed range operators and Moore-Penrose inverse

TL;DR

This paper investigates how perturbations by bounded operators affect the closedness of the range and the Moore-Penrose inverse of closed operators between Hilbert spaces, addressing stability and inverse relations.

Contribution

It provides conditions for the closedness of the range under perturbations and explores the relationships between Moore-Penrose inverses of perturbed and unperturbed operators.

Findings

Conditions ensuring the closedness of the range after perturbation

Relations between $T^{ ext{ extdagger}}$ and $(T+S)^{ ext{ extdagger}}$

Connections between $T^{ ext{ extdagger}}$ and $S^{ ext{ extdagger}}$

Abstract

Let $H_1,H_2$ be complex Hilbert spaces and $T:H_1\rightarrow H_2$ be a densely defined closed operator with domain $D(T)\subseteq H_1$ and $T^{\dagger}$ be the Moore-Penrose inverse of $T$. Let $S:H_1\rightarrow H_2$ be a bounded operator. In this article we focus our attention on the following questions: $1.$ Under what conditions closedness of range of $T$ will imply the closedness of range of $T+S$? $2.$ What is the relation between $T^{\dagger}$ and $(T+S)^{\dagger}$? $3.$ What is the relation between $T^{\dagger}$ and $S^{\dagger}$?.