Remarks on strictly singular operators

TL;DR

This paper explores conditions under which all bounded linear operators between certain Banach spaces are strictly singular, highlighting implications for operator theory and space structure.

Contribution

It provides new sufficient conditions and discusses consequences for the phenomenon where all bounded operators are strictly singular.

Findings

Identifies conditions for $LB(E,F)=L_s(E,F)$

Shows implications for the structure of Banach spaces

Highlights importance in operator theory

Abstract

A continuous linear operator $T:E \to F$ is called strictly singular if it cannot be invertible on any infinite dimensional closed subspace of its domain. In this note we discuss sufficient conditions and consequences of the phenomenon $LB(E,F)=L_s(E,F)$, which means that every continuous linear bounded operator defined on $E$ into $F$ is strictly singular.