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

## TL;DR

This paper investigates classes of operators between vector and normed lattices that preserve unbounded order and unbounded norm convergence, exploring their properties and relationships.

## Contribution

It introduces and analyzes unbounded $\sigma$-order-to-norm continuous and unbounded norm continuous operators, highlighting their properties and connections to existing operator classes.

## Key findings

- Characterization of unbounded $\sigma$-order-to-norm continuous operators.
- Relationship between unbounded norm continuous and other operator classes.
- Properties and structural results of these operators.

## Abstract

An operator $T $ from a vector lattice $E$ into a normed lattice $F$ is called unbounded $\sigma$-order-to-norm continuous whenever $x_{n}\xrightarrow{uo}0$ implies $\| Tx_{n}\|\rightarrow 0$, for each sequence $(x_{n})_n\subseteq E$. For a net $(x_{\alpha})_{\alpha}\subseteq E$, if $x_{\alpha}\xrightarrow{un}0$ implies $Tx_{\alpha}\xrightarrow{un}0$, 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.