### 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

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