Document Type : Research Paper

Authors

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

Department of Mathematics and Applications, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran

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 $x_{\alpha }\stackrel{Fo}{\to }0$ implies $T(x_{\alpha })\stackrel{\Vert .\Vert }{\to }0$ 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 $T:E\to X$ 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.

Keywords

• Vector lattice
• property $(F)$
• $\tilde{o}$-convergence
• order-to-norm continuous operator
• $\tilde{o}$rder-norm continuous operator
• $\tilde{o}$rder weakly compact