Weak precompactness in Banach lattices

TL;DR

This paper characterizes when the solid hull of weakly precompact sets in Banach lattices remains weakly precompact, linking it to properties of order intervals and disjoint sets, and explores domination of weakly precompact operators.

Contribution

It provides a characterization of weak precompactness preservation under solid hulls and investigates domination properties of weakly precompact positive operators in Banach lattices.

Findings

Solid hull of weakly precompact sets is weakly precompact under certain conditions.

Weakly precompact positive operators are dominated by weakly precompact operators under specific lattice conditions.

Characterization of when dominated positive operators are weakly precompact based on lattice properties.

Abstract

We show that the solid hull of every weakly precompact set of a Banach lattice $E$ is weakly precompact if and only if every order interval in $E$ is weakly precompact, or equivalently, if and only if every disjoint weakly compact set is weakly precompact. Some results on the domination property for weakly precompact positive operators are obtained. Among other things, we show that, for a pair of Banach lattices $E$ and $F$ with $E$ $\sigma$-Dedekind complete, every positive operator from $E$ to $F$ dominated by a weakly precompact operator is weakly precompact if and only if either the norm of $E^{\prime}$ is order continuous or else every order interval in $F$ is weakly precompact.