Multiplicative order compact operators between vector lattices and Riesz algebras

Abstract

In this paper, we present and examine the concept of multiplicative order compact operators from vector lattices from vector lattices to Riesz algebras. Specifically, a linear operator $T$ from a vector lattice $X$ to an Riesz algebra $E$ is deemed $\mathbb{omo}$-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 $y\in E$. Moreover, we introduce and scrutinize $\mathbb{omo}$-$M$- and $\mathbb{omo}$-$L$-weakly compact operators.