On weakly compact operators on Banach lattices

Abstract:

Consider a Banach lattice $E$ and an order bounded weakly compact operator $T:E \to E$. The purpose of this note is to study the weak compactness of operators that are related with $T$ in some order sense. The main results are the following. (1) If $T$ is a positive weakly compact operator and an operator $S:E \to E$ satisfies $0 \leqslant S \leqslant T$, then ${S^2}$ is weakly compact. (Examples show that $S$ need not be weakly compact.) (2) If $T$ and $S$ are as in (1) and either $S$ is an orthomorphism or $E$ has an order continuous norm, then $S$ is weakly compact. (3) If $E$ is an abstract $L$-space and $T$ is weakly compact, then the modulus $|T|$ is weakly compact.