Banach Space-Valued Extensions of Linear Operators on $L^{\mathrm{\infty }}$

Let $E$ and $G$ be two Banach function spaces, let $T\in \mathcal{L}(E,Y)$, and let $⟨X,Y⟩$ be a Banach dual pair. In this paper we give conditions for which there exists a (necessarily unique) bounded linear operator $T_Y\in \mathcal{L}(E(Y),G(Y))$ with the property that $$⟨x,T_{Y}e⟩=T⟨x,e⟩,e\in E(Y),x\in X\mathrm{.}$$ Our first main result states that, in case ${\langle X,Y \rangle} = {\langle Y^{*}, Y \rangle}$with$Y$areflexiveBanachspace,fortheexistenceof$T_{Y}$itsufficientthat$T$ is dominated by a positive operator. Our second main result concerns the case that $T$ is an adjoint operator on $L^{\mathrm{\infty }}(A)$: we suppose that $E=L^{\mathrm{\infty }}(A)$ for a semi-finite measure space $(A,\mathcal{A},\mu )$, that $⟨F,G⟩$ is a K"othe dual pair, and that $T$ is $\sigma (L^{\mathrm{\infty }}(A),L^1(A))$-to-$\sigma (G,F)$ continuous. Then $T_Y$ exists provided that $T$ is dominated by a positive operator, in which case $T_Y$ is $\sigma (L^{\mathrm{\infty }}(A;Y),L^1(A;X))$-to-$\sigma (G(Y),F\tilde{\otimes }X)$ continuous; here $F\tilde{\otimes }X$ denotes the closure of $F\otimes X$ in $F(X)$. We also consider situations in which the existence is automatic and we furthermore show that in certain situations it is necessary that $T$ is regular. As an application of this result we consider conditional expectation on Banach space-valued $L^{\mathrm{\infty }}$-spaces.