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 $T:E\rightarrow F$ between two Riesz spaces is said to be strongly order continuous (resp. strongly $\sigma$-order continuous), if $x _\alpha \xrightarrow{uo}0$ (resp. $x _n \xrightarrow{uo}0$) in $E$ implies $Tx _\alpha \xrightarrow{o}0$ (resp. $Tx _n \xrightarrow{o}0$) 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 $E^\sim$.