$DW$-compact operators on Banach lattices

TL;DR

This paper characterizes $DW$-compact operators on Banach lattices as those that are both Dunford-Pettis and $AM$-compact, providing insights into their structure and applications in Banach lattice theory.

Contribution

It establishes a precise characterization of $DW$-compact operators as the intersection of Dunford-Pettis and $AM$-compact operators, and applies this to Banach lattice properties.

Findings

$DW$-compact operators are exactly the Dunford-Pettis and $AM$-compact operators.

Characterization of Banach lattices where disjointly weakly compact sets are limited or Dunford-Pettis.

Application of $DW$-compact operators to identify specific Banach lattice properties.

Abstract

This paper is devoted to the study of $DW$-compact operators, that is, those operators which map disjointly weakly compact sets in a Banach lattice onto relatively compact sets. We show that $DW$-compact operators are precisely the operators which are both Dunford-Pettis and $AM$-compact. As an application, Banach lattices with the property that every disjointly weakly compact set is a limited (resp. Dunford-Pettis) set, are characterized by using $DW$-compact operators.