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.