Totally bounded sets in the absolute weak topology

TL;DR

This paper characterizes totally bounded sets in the absolute weak topology of Banach lattices using almost Dunford-Pettis operators and explores conditions under which positive operators are PL-compact.

Contribution

It introduces a new characterization of totally bounded sets in the absolute weak topology via almost Dunford-Pettis operators and applies this to positive operators and PL-compactness.

Findings

A bounded set is totally bounded in the absolute weak topology iff its image under certain operators is relatively compact.

Positive operators dominated by PL-compact operators are PL-compact under specific conditions.

The norm of the dual space being order continuous influences operator compactness properties.

Abstract

In this paper, almost Dunford-Pettis operators with ranges in $c_0$ are used to identify totally bounded sets in the absolute weak topology. That is, a bounded subset $A$ of a Banach lattice $E$ is $|\sigma|(E,E^\prime)$-totally bounded if and only if $T(A)\subset c_0$ is relatively compact for every almost Dunford-Pettis operator $T:E\to{c_0}$. As an application, we show that for two Banach lattices $E$ and $F$ every positive operator from $E$ to $F$ dominated by a PL-compact operator is PL-compact if and only if either the norm of $E^{\,\prime}$ is order continuous or every order interval in $F$ is $|\sigma|(F,F^\prime)$-totally bounded.