Unbounded convergence in Banach lattices and applications

TL;DR

This paper unifies various types of unbounded convergence in Banach lattices, providing characterizations of weakly compact sets and operators, and advancing the understanding of convergence behaviors in these mathematical structures.

Contribution

It introduces a comprehensive framework for all unbounded convergences in Banach lattices and characterizes related compactness properties, which was not previously established.

Findings

Characterization of L-weakly compact sets

Characterization of L-weakly compact operators

Characterization of M-weakly compact operators

Abstract

Several recent papers investigated unbounded convergences in Banach lattices. Combine all unbounded convergences, including unbounded order (norm, absolute weak, absolute weak*) convergence, we characterize L-weakly compact sets, L-weakly compact operators and M-weakly compact operators on Banach lattices. Some related results are obtained as well.