Unbounded $\sigma$-order-to-norm continuous and $un$-continuous operators

Abstract

An operator $T$ from a vector lattice $E$ into a normed lattice $F$ is called unbounded $\sigma$-order-to-norm continuous whenever $x_n\stackrel{uo}{\to }0$ implies $\Vert T{x}_n\Vert \to 0$, for each sequence $(x_n{)}_n\subseteq E$. For a net $(x_{\alpha }{)}_{\alpha }\subseteq E$, if $x_{\alpha }\stackrel{un}{\to }0$ implies $T{x}_{\alpha }\stackrel{un}{\to }0$, then $T$ is called an unbounded norm continuous operator. In this manuscript, we study some properties of these classes of operators and their relationships with the other classes of operators.