Strongly order continuous operators on Riesz spaces

TL;DR

This paper introduces strongly order continuous and strongly σ-order continuous operators on Riesz spaces, exploring their properties, conditions for equivalence with order continuity, and the structure of their functionals.

Contribution

It defines new classes of operators on Riesz spaces and investigates their properties and relationships with existing concepts like order continuity.

Findings

Strongly order continuous operators are characterized by uo-convergence implying order convergence.

Conditions are provided under which order continuity and strong order continuity coincide.

The set of all so-continuous linear functionals forms a band in the dual space.

Abstract

In this paper we introduce two new classes of operators that we call strongly order continuous and strongly $\sigma$-order continuous operators. An operator $T:E\rightarrow F$ between two Riesz spaces is said to be strongly order continuous (resp. strongly $\sigma$-order continuous), if $x _\alpha \xrightarrow{uo}0$ (resp. $x _n \xrightarrow{uo}0$) in $E$ implies $Tx _\alpha \xrightarrow{o}0$ (resp. $Tx _n \xrightarrow{o}0$) in $F$. We give some conditions under which order continuity will be equivalent to strongly order continuity of operators on Riesz spaces. We show that the collection of all $so$-continuous linear functionals on a Riesz space $E$ is a band of $E^\sim$.