On unbounded order continuous operators

Abstract

Let $U$ and $V$ be two Archimedean Riesz spaces. An operator $S:U\rightarrow V$ is said to be unbounded order continuous ($uo$-continuous), if $r_{\alpha }\overset{uo}{\rightarrow }0$ in $U$ implies $Sr_{\alpha }\overset{uo}{% \rightarrow }0$ in $V$. In this paper, we give some properties of the $uo$% -continuous dual $U_{uo}^{\sim }$ of $U$. We show that a nonzero linear functional $f$ on $U$ is $uo$-continuous if and only if $f$ is a linear combination of finitely many order continuous lattice homomorphisms. The result allows us to characterize the $uo$-continuous dual $U_{uo}^{\sim }.$ In general, by giving an example that the $uo$-continuous dual $U_{uo}^{\sim }$ is not a band in $U^{\sim }$, we obtain the conditions for the $uo$% -continuous dual of a Banach lattice $U$ to be a band in $U^{\sim }$. Then, we examine the properties of $uo$-continuous operators. We show that $S$ is an order continuous operator if and only if $S$ is an unbounded order continuous operator when $S$ is a lattice homomorphism between two Riesz spaces $U$ and $V$. Finally, we proved that if an order bounded operator $S:U\rightarrow V$ between Archimedean Riesz space $U$ and atomic Dedekind complete Riesz space $V$ is $uo$-continuous, then $\left\vert S\right\vert $ is $uo$-continuous.