Unbounded order-norm continuous and unbounded norm continuous operators

Abstract

A continuous operator $T$ between two normed vector lattices $E$ and
$F$ is called unbounded order-norm continuous
whenever $x_{\alpha}\xrightarrow{uo}0$ implies
$\| Tx_{\alpha}\|\rightarrow 0$,
for each norm bounded net $(x_{\alpha})_\alpha\subseteq E$. Let $E$ and $F$ be two Banach lattices. A continuous operator $T:E\rightarrow F$ is called unbounded norm continuous,
if for each norm bounded net $(x_{\alpha})_{\alpha}\subseteq E$, $x_{\alpha}\xrightarrow{un}0$ implies
$Tx_{\alpha}\xrightarrow{un}0$.
In this manuscript, we study some
properties of these classes of operators and their relationships
with the other classes of operators.
\keywords{ unbounded $\sigma$-order-norm continuous \and unbounded order-norm continuous \and $\sigma$-unbounded norm continuous \and unbounded norm continuous \and $un$-compact