Quantifications of strictly singular operators and strictly cosingular operators

TL;DR

This paper explores quantitative measures of various classes of operators, including strictly singular and cosingular operators, and establishes strengthened relationships among these classes and other operator types.

Contribution

It introduces quantitative versions of known relationships among strictly singular, cosingular, compact, and weakly compact operators, enhancing the understanding of their interconnections.

Findings

Quantitative relationships between strictly singular and compact operators.

Strengthened versions of classical theorems relating these operator classes.

New insights into the structure of strictly cosingular operators.

Abstract

We investigate possible quantifications of strictly singular operators, $l_{p}$-strictly singular operators, $c_{0}$-strictly singular operators, strictly cosingular operators, $l_{p}$-strictly cosingular operators. We prove quantitative, even strengthening versions of well-known results about relationships of these five classes of operators and compact, weakly compact, unconditionally converging operators.