Extension of order bounded operators

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 $T^t\in \mathcal{L}_b(E^t,F)$.