The perturbation classes problem for subprojective and superprojective Banach spaces

TL;DR

This paper characterizes the perturbation classes for semi-Fredholm operators between Banach spaces, showing they coincide with strictly singular or strictly cosingular operators under subprojectivity or superprojectivity assumptions.

Contribution

It extends previous results by establishing these perturbation class characterizations under weaker conditions on Banach spaces.

Findings

Perturbation class for upper semi-Fredholm operators equals strictly singular operators when X is subprojective.

Perturbation class for lower semi-Fredholm operators equals strictly cosingular operators when Y is superprojective.

Results generalize earlier work with stronger assumptions.

Abstract

We show that the perturbation class for the upper semi-Fredholm operators between two Banach spaces X and Y coincides with the strictly singular operators when X is subprojective and that the perturbation class for the lower semi-Fredholm operators coincides with the strictly cosingular operators when Y is superprojective. Similar results were previously obtained under stronger conditions for X and Y.