Banach Space-valued Extensions of Linear Operators on L^∞

Abstract

Let E and G be two Banach function spaces, let T ∈ L(E, Y ), and let $\begin{array}{lll} \left\langle {X,\,Y} \right\rangle \end{array}$ be a Banach dual pair. In this paper we give conditions for which there exists a (necessarily unique) bounded linear operator TY ∈ L(E(Y), G(Y )) with the property that

$$ \begin{array}{lll} \left\langle {x,\;Tye} \right\rangle = T\left\langle {x,\,e} \right\rangle ,\,e \in E(Y),\,x \in X. \end{array} $$

The first main result states that, in case $\begin{array}{lll} \left\langle {X,\,Y} \right\rangle = \left\langle {Y^* ,\,Y} \right\rangle \end{array}$ with Y a reflexive Banach space, for the existence of TY it sufficient that T is dominated by a positive operator. We furthermore show that for Y within a wide class of Banach spaces (including the Banach lattices) the validity of this extension result for E = l^∞ and G = K even characterizes the reflexivity of Y . The second main result concerns the case that T is an adjoint operator on L^∞(A): we assume that E = L^∞(A) for a semi-finite measure space (A, A, μ), that $\begin{array}{lll} \left\langle {F,\,G} \right\rangle \end{array}$ is a Köthe dual pair, and that T is σ(L ^∞ (A),L1(A))- to-σ(G, F) continuous. In this situation we show that TY also exists provided that T is dominated by a positive operator. As an application of this result we consider conditional expectation on Banach space-valued L ^∞ -spaces.