$\tilde{o}$rder-norm continuous operators and $\tilde{o}$rder weakly compact operators

TL;DR

This paper introduces and studies $ ilde{o}$rder-norm continuous and $ ilde{o}$rder weakly compact operators, exploring their properties and relationships within vector lattice theory.

Contribution

It defines new classes of operators based on $ ilde{o}$rder convergence and investigates their properties and connections with existing operator classifications.

Findings

Characterization of $ ilde{o}$rder-norm continuous operators.

Introduction of $ ilde{o}$rder weakly compact operators.

Relationships between new and known operator classes.

Abstract

Let $E$ be a sublattice of a vector lattice $F$. A continuous operator $T$ from the vector lattice $E$ into a normed vector space $X$ is said to be $\tilde{o}$rder-norm continuous whenever $x_\alpha\xrightarrow{Fo}0$ implies $Tx_\alpha\xrightarrow{\Vert.\Vert}0$ for each $(x_\alpha)_\alpha\subseteq E$. Our mean from the convergence $ x_\alpha\stackrel{Fo} {\longrightarrow} x $ is that there exists another net $ \left(y_\alpha\right) $ in $F $ with the same index set satisfying $ y_\alpha\downarrow 0 $ in $F$ and $ \vert x_\alpha - x \vert \leq y_\alpha $ for all indexes $ \alpha $. In this paper, we will study some properties of this new class of operators and its relationships with some known classifications of operators. We also define the new class of operators that named $\tilde{o}$rder weakly compact operators. A continuous operator $T: E \rightarrow X $ is said to be $\tilde{o}$rder weakly compact, if $ T(A) $ in $X$ is a relatively weakly compact set for each $Fo$-bounded $A\subseteq E$. In this manuscript, we study some properties of this class of operators and its relationships with $\tilde{o}$rder-norm continuous operators.