A generalization of order continuous operators

Abstract

Let $E$ be a sublattice of a vector lattice $F$. A net $\{x_{\alpha }{\}}_{\alpha \in \mathcal{A}}\subseteq E$ is said to be $F$-order convergent to a vector $x\in E$ (in symbols $x_{\alpha }\stackrel{Fo}{\to }x$ in $E$), whenever there exists a net $\{y_{\beta }{\}}_{\beta \in \mathcal{B}}$ in $F$ satisfying $y_{\beta }\downarrow 0$ in $F$ and for each $\beta$, there exists ${\alpha }_0$ such that $|x_{\alpha }-x|\le y_{\beta }$ whenever $\alpha \ge {\alpha }_0$. In this manuscript, first we study some properties of $F$-order convergence nets and we extend some results to the general cases. Let $E$ and $G$ be sublattices of vector lattices $F$ and $H$ respectively. We introduce $FH$-order continuous operators, that is, an operator $T$ between two vector lattices $E$ and $G$ is said to be $FH$-order continuous, if $x_{\alpha }\stackrel{Fo}{\to }0$ in $E$ implies $T{x}_{\alpha }\stackrel{Ho}{\to }0$ in $G$. We will study some properties of this new classification of operators and its relationships with order continuous operators.

arXiv admin note: text overlap with arXiv:1908.03193