Weak compactness of almost limited operators

TL;DR

This paper investigates the conditions under which almost limited operators between Banach lattices are weakly compact, establishing equivalences involving reflexivity and order continuity of the norm.

Contribution

It provides new characterizations of weak compactness for almost limited operators in terms of reflexivity and order continuity in Banach lattices.

Findings

Almost limited operators are weakly compact iff E is reflexive or F has order continuous norm.

The square of positive almost limited operators is weakly compact iff E's norm is order continuous.

Results connect operator properties with lattice-theoretic conditions.

Abstract

The paper is devoted to the relationship between almost limited operators and weakly compacts operators. We show that if $F$ is a $\sigma $-Dedekind complete Banach lattice then, every almost limited operator $T:E\rightarrow F $ is weakly compact if and only if $E$ is reflexive or the norm of $F$ is order continuous. Also, we show that if $E$ is a $\sigma $-Dedekind complete Banach lattice then the square of every positive almost limited operator $ T:E\rightarrow E$ is weakly compact if and only if the norm of $E$ is order continuous.