Unbounded continuous operators and unbounded Banach-Saks property in Banach lattices

TL;DR

This paper introduces unbounded continuous operators in Banach lattices using $uaw$-convergence, characterizes key lattice properties via these operators, and explores an unbounded Banach-Saks property with various relations to classical properties.

Contribution

It defines unbounded continuous operators using $uaw$-convergence and characterizes Banach lattice properties through these operators and the unbounded Banach-Saks property.

Findings

Characterization of order continuous Banach lattices via unbounded operators.

Characterization of reflexive Banach lattices using $uaw$-Cauchy sequences.

Introduction of the unbounded Banach-Saks property and its relations to classical properties.

Abstract

Motivated by the equivalent definition of a continuous operator between Banach spaces in terms of weakly null nets, we introduce unbounded continuous operators by replacing weak convergence with the unbounded absolutely weak convergence ( $uaw$-convergence) in the definition of a continuous operator between Banach lattices. We characterize order continuous Banach lattices and reflexive Banach lattices in terms of these spaces of operators. Moreover, motivated by characterizing of a reflexive Banach lattice in terms of unbounded absolutely weakly Cauchy sequences, we consider pre-unbounded operators between Banach lattices which maps $uaw$-Cauchy sequences to weakly ( $uaw$- or norm) convergent sequences. This allows us to characterize $KB$-spaces and reflexive spaces in terms of these operators, too. Furthermore, we consider the unbounded Banach-Saks property as an unbounded version of the weak Banach-Saks property. There are many considerable relations between spaces possessing the unbounded Banach-Saks property with spaces fulfilled by different types of the known Banach-Saks property. In particular, we characterize order continuous Banach lattices in terms of these relations, as well.