Multiplicative order compact operators between vector lattices and $l$-algebras

TL;DR

This paper introduces and studies multiplicative order compact operators from vector lattices to $l$-algebras, expanding the understanding of operator compactness in ordered algebraic structures.

Contribution

It defines new classes of multiplicative order compact operators and explores their properties within vector lattices and $l$-algebras.

Findings

Introduction of $ ext{ extit{omo}}$-compact operators

Development of $ ext{ extit{omo}}$-$M$- and $ ext{ extit{omo}}$-$L$-weakly compact operators

Analysis of properties and relationships of these operators

Abstract

In the present paper, we introduce and investigate the multiplicative order compact operators from vector lattices to $l$-algebras. A linear operator $T$ from a vector lattice $X$ to an $l$-algebra $E$ is said to be $\mathbb{omo}$-compact if every order bounded net $x_\alpha$ in $X$ possesses a subnet $x_{\alpha_\beta}$ such that $Tx_{\alpha_\beta}\moc y$ for some $y\in E$. We also introduce and study $\mathbb{omo}$-$M$- and $\mathbb{omo}$-$L$-weakly compact operators from vector lattices to $l$-algebras.