Weak Compactness of Almost Limited Operators

摘要

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