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

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Ghanizadeh Zare Sajjad; Haghnejad Azar Kazem; Matin Mina; Hazrati Somayeh

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Journal Paper

Paper Information

Journal: INTERNATIONAL JOURNAL OF NONLINEAR ANALYSIS AND APPLICATIONS Year:2025 | Volume:16 | Issue:3 Page(s): 167-174

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Information Journal Paper

Title

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

Author(s)

Ghanizadeh Zare Sajjad | Haghnejad Azar Kazem | Matin Mina | Hazrati Somayeh | Issue Writer Certificate

Keywords

Vector lattice

property $(F)$

$\tilde{o}$-convergence

order-to-norm continuous operator

$\tilde{o}$rder-norm continuous operator

$\tilde{o}$rder weakly compact

Abstract

Let $E$ be a sublattice of a Vector lattice $F$. A continuous operator $T$ from $E$ into a normed vector space $X$ is said to be $\tilde{o}$rder-norm continuous if $x_\alpha\xrightarrow{Fo}0$ implies $T(x_\alpha)\xrightarrow{\Vert.\Vert}0$ for every $(x_{\alpha})_{\alpha \in A}\subseteq E$. This paper aims to investigate the properties of this new class of operators and explore their relationships with existing classifications of operators. We introduce a new class of operators called $\tilde{o}$rder weakly compact operators. A continuous operator $T: E \rightarrow X $ is considered $\tilde{o}$rder weakly compact if $ T(A) $ in $X$ is a relatively weakly compact set for every $Fo$-bounded $A\subseteq E$. In this manuscript, we examine various properties of this class of operators and explore their connections with $\tilde{o}$rder-norm continuous operators.

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