Multilinear operators on spaces of continuous functions

Abstract

Let E1, . . . ,Ed be Banach spaces such that all linear operators from Ei into E_j (i 6= j) are weakly compact. The authors show that every continuous d-linear operator T on E1 × • • • × Ed to a Banach space F possesses a unique bounded multilinear extension T__ : E__ 1 × • • • × E__ d ! F__ that is !_ − !_-separately continuous and kT__k = kTk. In particular, existence of unique continuous multilinear extensions from C(K1)ו • •× C(Kd) (Ki – Hausdorff compact spaces) to C(K1)__ו • •×C(Kd)__ that are separately weak_-continuous is established. As a corollary, integral representations with respect to polymeasures for multilinear mappings on C(K1)ו • •×C(Kd) into a Banach space are found. The results generalize a theorem due to Pelczynsky about multilinear extensions from C(K1) × • • • × C(Kd) to the Cartesian product of the spaces of bounded Baire functions on Ki.