# Extension of order bounded operators

## TL;DR

This paper develops an extension norm for order dense majorizing sublattices of vector lattices and explores how lattice and topological properties extend to these larger spaces, especially for operators into Dedekind complete Banach lattices.

## Contribution

It introduces an extension norm for order dense majorizing sublattices and studies the extension of lattice and topological properties of operators into Dedekind complete Banach lattices.

## Key findings

- Extension norm $\
- $T^t$ preserves operator norm and lattice homomorphism properties.
- Positive operators maintain their norm under extension, and $T^t$ remains a lattice homomorphism.

## Abstract

Assume that a normed lattice $E$ is order dense majorizing of a vector lattice $E^t$. There is an extension norm $\Vert.\Vert_t$ for $E^t$ and we extend some lattice and topological properties of normed lattice $(E,\Vert.\Vert)$ to new normed lattice $(E^t,\Vert.\Vert_t$). For a Dedekind complete Banach lattice $F$, $T^t$ is an extension of $T$ from $E^t$ into $F$ whenever $T$ is an order bounded operator from $E$ into $F$. For each positive operator $T$, we have $\Vert T\Vert=\Vert T^t\Vert$ and we show that $T^t$ is a lattice homomorphism from $E^t$ into $F$ and moreover $T^t\in \mathcal{L}_n(E^t,F)$ whenever $0\leq T\in \mathcal{L}_n(E,F)$ and $T(x\wedge y)=Tx \wedge Ty$ for each $0\leq x,y\in E$. We also extend some lattice and topological properties of $T\in \mathcal{L}_b(E,F)$ to the extension operator $T^t\in \mathcal{L}_b(E^t,F)$.