# Quasi-Fredholm spectrum and compact perturbations

## TL;DR

This paper investigates the properties of the quasi-Fredholm spectrum of operators on Banach spaces and examines the stability of the SVEP under compact perturbations in Hilbert spaces.

## Contribution

It introduces new insights into the quasi-Fredholm spectrum and characterizes operators preserving the SVEP under compact perturbations.

## Key findings

- Characterization of the quasi-Fredholm resolvent set.
- Conditions for SVEP stability under compact perturbations.
- Identification of operators maintaining SVEP in Hilbert spaces.

## Abstract

In this paper we explore some characteristics of the quasi-Fredholm resolvent set $\rho_{qf}(T)$ of an operator $T$ defined on an infinite dimensional Banach space $X$. Moreover, in the case of Hilbert space $H$, we study the stability of the SVEP and describe the operators for which the SVEP is preserved under compact perturbations using quasi-Fredholm spectrum and $\rho_{qf}(T)$.