Descriptive set theoretic methods applied to strictly singular and strictly cosingular operators

TL;DR

This paper explores the structure of strictly singular and cosingular operators in Banach spaces, using set theoretic methods and Schreier sets to analyze their classification and address a question posed by J. Diestel.

Contribution

It introduces a set theoretic framework to decompose strictly singular operators and investigates whether similar decompositions apply to strictly cosingular operators.

Findings

Strictly singular operators form an increasing union of $ ext{ω}_1$ subclasses defined via Schreier sets.

The paper addresses Diestel's question regarding the analogous structure for strictly cosingular operators.

Abstract

The class of strictly singular operators originating from the dual of a separable Banach space is written as an increasing union of $\omega_1$ subclasses which are defined using the Schreier sets. A question of J. Diestel, of whether a similar result can be stated for strictly cosingular operators, is studied.