Some characterizations of almost limited sets and applications

TL;DR

This paper characterizes almost limited sets and operators in Banach lattices, extending previous results, and explores their relationships with other classes of operators, providing new insights into the structure of Banach lattices.

Contribution

It introduces and studies the class of almost limited operators and generalizes existing results on almost limited sets in Banach lattices.

Findings

Characterizations of almost limited sets in Banach lattices.

Introduction of almost limited operators and their properties.

Relationships between almost limited operators and other operator classes.

Abstract

Recently, J.X. Chen et al. introduced and studied the class of almost limited sets in Banach lattices. In this paper we establish some characterizations of almost limited sets in Banach lattices (resp. wDP* property of Banach lattices), that generalize some results obtained by J.X. Chen et al.. Also, we introduce and study the class of the almost limited operators, which maps the closed unit bull of a Banach space to an almost limited subset of a Banach lattice. Some results about the relationship between the class of almost limited operators and that of limited (resp. L- and M-weakly compact, resp. compact) operators are presented.