Strongly order continuous operators on Riesz spaces

Abstract

In this paper we introduce two new classes of operators that we call strongly order continuous and strongly $\sigma$-order continuous operators. An operator $T:E\to F$ between two Riesz spaces is said to be strongly order continuous (resp. strongly $\sigma$-order continuous), if $x_{\alpha }\stackrel{uo}{\to }0$ (resp. $x_n\stackrel{uo}{\to }0$) in $E$ implies $T{x}_{\alpha }\stackrel{o}{\to }0$ (resp. $T{x}_n\stackrel{o}{\to }0$) in $F$. We give some conditions under which order continuity will be equivalent to strongly order continuity of operators on Riesz spaces. We show that the collection of all $so$-continuous linear functionals on a Riesz space $E$ is a band of $E^{\sim }$.

References in corpus (1)

• Uo-convergence and its applications to Cesàro means in Banach lattices