Multilinear operators on spaces of continuous functions
Let \(E_1, \dots, E_d\) be Banach spaces such that all linear operators from \(E_i\) into \(E_j^*\) (\(i\neq j\)) are weakly compact. The authors show that every continuous \(d\)-linear operator \(T\) on \(E_1\times\dots\times E_d\) to a Banach space \(F\) possesses a unique bounded multilinear extension \(T^{**}: E_1^{**}\times\dots\times E_d^{**} \to F^{**}\) that is \(\omega^*-\omega^*\)-separately continuous and \(\| T^{**}\| =\| T\| \). In particular, existence of unique continuous multilinear extensions from \(C(K_1)\times\dots\times C(K_d)\) (\(K_i\) -- Hausdorff compact spaces) to \(C(K_1)^{**}\times\dots\times C(K_d)^{**}\) that are separately weak\(^*\)-continuous is established. As a corollary, integral representations with respect to polymeasures for multilinear mappings on \(C(K_1)\times\dots\times C(K_d)\) into a Banach space are found. The results generalize a theorem due to Pelczynsky about multilinear extensions from \(C(K_1)\times\dots\times C(K_d)\) to the Cartesian product of the spaces of bounded Baire functions on \(K_i\).
scientific article; zbMATH DE number 1251476