Banach Space-valued Extensions of Linear Operators on L∞

Let E and G be two Banach function spaces, let T is an element of 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 is an element of L(E(Y), G(Y)) with the property that < x, T(Y)e > = T < x, e >, e is an element of E(Y), x is an element of X. The first main result states that, in case < X, Y > = < Y*, Y > with Y a reflexive Banach space, for the existence of T-Y 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(infinity) and G = K even characterizes the reflexivity of Y. The second main result concerns the case that T is an adjoint operator on L-infinity(A): we assume that E = L-infinity(A) for a semi-finite measure space (A, A, mu), that < F, G > is a Kothe dual pair, and that T is sigma(L-infinity(A), L-1 (A))to-sigma(G, F) continuous. In this situation we show that T-Y 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-infinity- spaces.