Order-to-topology continuous operators

Abstract

An operator $T$ from vector lattice $E$ into vector topology $(F,\tau )$ is said to be order-to-topology continuous whenever $x_{\alpha }\stackrel{o}{\to }0$ implies $T{x}_{\alpha }\stackrel{\tau }{\to }0$ for each $(x_{\alpha }{)}_{\alpha }\subset E$. The collection of all order-to-topology continuous operators will be denoted by $L_{o\tau }(E,F)$. In this paper, we will study some properties of this new classification of operators. We will investigate the relationships between order-to-topology continuous operators and others classes of operators such as order continuous, order weakly compact and $b$-weakly compact operators.