$DW$-DP operators and $DW$-limited operators on Banach lattices

TL;DR

This paper introduces and characterizes two new classes of operators on Banach lattices, called $DW$-DP and $DW$-limited operators, and explores their properties and approximation behaviors.

Contribution

It defines $DW$-DP and $DW$-limited operators, characterizes them via existing operator classes, and investigates their approximation properties.

Findings

$DW$-DP operators are exactly the weak and order Dunford-Pettis operators.

$DW$-limited operators are exactly the weak$^*$ Dunford-Pettis and order limited operators.

Positive $DW$-DP and $DW$-limited operators have specific approximation properties.

Abstract

This paper is devoted to the study of two classes of operators related to disjointly weakly compact sets, which we call $DW$-DP operators and $DW$-limited operators, respectively. They carry disjointly weakly compact subsets of a Banach lattice onto Dunford-Pettis sets and limited sets, respectively. We show that $DW$-DP (resp. $DW$-limited) operators are precisely the operators which are both weak Dunford-Pettis and order Dunford-Pettis (resp. weak$^*$ Dunford-Pettis and order limited) operators. Furthermore, the approximation properties of positive $DW$-DP and positive $DW$-limited operators are given.