Order-to-topology continuous operators

Abstract

An operator T from vector lattice E into topological vector space $(F,\tau )$ is said to be order-to-topology continuous whenever $x_\alpha \xrightarrow {o}0$ implies $Tx_\alpha \xrightarrow {\tau }0$ 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 class 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. Under some sufficient and necessary conditions we show that the adjoint of order-to-norm continuous operators is also order-to-norm continuous and vice verse.