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

Abstract

Let $E$ and $G$ be two Banach function spaces, let $T \in \mathcal{L}(E,Y)$, and let ${\langle X,Y \rangle}$ 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 \[ {\langle x,T_{Y}e \rangle} = T{\langle x,e \rangle}, \quad\quad\quad e \in E(Y), x \in X. \] Our first main result states that, in case ${\langle X,Y \rangle} = {\langle Y^{*}, Y \rangle}$ with $Y$ a reflexive Banach space, for the existence of $T_{Y}$ it sufficient that $T$ is dominated by a positive operator. Our second main result concerns the case that $T$ is an adjoint operator on $L^{\infty}(A)$: we suppose that $E = L^{\infty}(A)$ for a semi-finite measure space $(A,\mathscr{A},\mu)$, that ${\langle F, G \rangle}$ is a K\"othe dual pair, and that $T$ is $\sigma(L^{\infty}(A),L^{1}(A))$-to-$\sigma(G,F)$ continuous. Then $T_{Y}$ exists provided that $T$ is dominated by a positive operator, in which case $T_{Y}$ is $\sigma(L^{\infty}(A;Y),L^{1}(A;X))$-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 $L^{\infty}$-spaces.