Order-to-topology continuous operators

TL;DR

This paper introduces and studies a new class of operators called order-to-topology continuous operators, analyzing their properties and relationships with existing classes like order continuous and weakly compact operators.

Contribution

It defines the class of order-to-topology continuous operators and explores their properties and connections with other well-known operator classes.

Findings

Characterization of order-to-topology continuous operators

Relationships with order continuous and weakly compact operators

Properties and structural results of the new operator class

Abstract

An operator $T$ from vector lattice $E$ into vector topology $(F,\tau)$ is said to be order-to-topology continuous whenever $x_\alpha\xrightarrow{o}0$ implies $Tx_\alpha\xrightarrow{\tau}0$ for each $(x_\alpha)_\alpha\subset E$. The collection of all order-to-topology continuous operators will be denoted by $L_{o\tau}(E,F)$. In this paper, we will study some properties of this new classification of operators. We will investigate the relationships between order-to-topology continuous operators and others classes of operators such as order continuous, order weakly compact and $b$-weakly compact operators.