Banach Space-valued Extensions of Linear Operators on L∞

Let E and G be two Banach function spaces, let T ∈ 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 TY ∈ L(E(Y), G(Y )) with the property that ⟨x,Tye⟩=T⟨x,e⟩,e∈E(Y),x∈X.
The first main result states that, in case ⟨X,Y⟩=⟨Y∗,Y⟩ 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 ⟨F,G⟩ 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.