Unbounded Banach-Saks operators and unbounded Grothendieck operators on Banach lattices

TL;DR

This paper explores unbounded Banach-Saks and Grothendieck operators on Banach lattices, providing new characterizations of lattice properties like order continuity and reflexivity through these operators.

Contribution

It introduces new classes of unbounded operators related to Banach-Saks and Grothendieck properties and characterizes key lattice properties via these operators.

Findings

Characterization of order continuity of Banach lattices.

Characterization of reflexivity of Banach lattices.

Extension of Banach-Saks and Grothendieck properties to unbounded operators.

Abstract

Suppose $E$ is a Banach lattice. Recently, there have been some motivating contexts regarding the known Banach-Saks property and the Grothendieck property from an order point of view. In this paper, we establish these results for operators that enjoy different types considered for the Banach-Saks property as well as for different notions related to the Grothendieck property. In particular, beside other results, we characterize order continuity and reflexivity of Banach lattices in terms of the corresponding bounded operators defined on them.