On unbounded order continuous operators 2

Abstract

Let E and F be two Archimedean Riesz spaces. An operator $T:E\rightarrow F$ is said to be unbounded order continuous (uo-continuous), if $u_{\alpha }\overset{uo}{\rightarrow }0$ in E implies $Tu_{\alpha }\overset{uo}{ \rightarrow }0$ in F. In this study, our main aim is to give the solution to two open problems which are posed by Bahramnezhad and Azar. Using this, we obtain that the space $L_{uo}(E,F)$ of order bounded unbounded order continuous operators is an ideal in $L_{b}(E,F)$ for Dedekind complete Riesz space F. In general, by giving an example that the space $L_{uo}(E,F)$ of order bounded unbounded order continuous operators is not a band in $L_{b}(E,F)$, we obtain the conditions on E or F for the space $L_{uo}(E,F)$ to be a band in $L_{b}(E,F)$. Then, we give the extension theorem for uo-continuous operators similar to Veksler’s theorem for order continuous operators.