Two classes of operators related to the perturbation classes problem

TL;DR

This paper investigates two classes of operators related to the perturbation classes problem for semi-Fredholm operators, providing conditions for equalities and solutions to the problem.

Contribution

It introduces and analyzes classes of operators between Banach spaces that relate to perturbation classes, offering new conditions for their equalities and solutions.

Findings

Identifies conditions under which operator classes coincide with perturbation classes.

Provides positive solutions to the perturbation classes problem for semi-Fredholm operators.

Establishes inclusion relations between operator classes and perturbation classes.

Abstract

Let $SS$ and $SC$ be the strictly singular and the strictly cosingular operators acting between Banach spaces, and let $P\Phi_+$ and $P\Phi_+$ be the perturbation classes for the upper and the lower semi-Fredholm operators. We study two classes of operators $\Phi S$ and $\Phi C$ that satisfy $SS\subset \Phi S \subset P\Phi_+$ and $SC\subset \Phi C\subset P\Phi_-$. We give some conditions under which these inclusions become equalities, from which we derive some positive solutions to the perturbation classes problem for semi-Fredholm operators.