Some properties on the unbounded absolute weak convergence in Banach lattices

TL;DR

This paper explores the relationships between unbounded absolute weak convergence, weak convergence, and operator properties in Banach lattices, providing characterizations and new insights into their interplay.

Contribution

It offers new characterizations of Banach lattices where certain null sequences are equivalent under different convergence modes and studies operator compactness properties.

Findings

Characterization of Banach lattices where weak null sequences are uaw-null.

Conditions under which un-null nets are weakly null in order continuous Banach lattices.

New characterization of b-weakly compact operators using uaw-convergence.

Abstract

In this paper, we investigate more about relationship between $uaw$ -convergence (resp. $un$-convergence) and the weak convergence. More precisely, we characterize Banach lattices on which every weak null sequence is $uaw$-null. Also, we characterize order continuous Banach lattices under which every norm bounded $un$-null net (resp. sequence) is weakly null. As a consequence, we study relationship between sequentially $uaw$-compact operators and weakly compact operators. Also, it is proved that every continuous operator, from a Banach lattice $E$ into a non-zero Banach space $X$, is unbounded continuous if and only if $E^{\prime }$ is order continuous. Finally, we give a new characterization of $b$-weakly compact operators using the $uaw$-convergence sequences.