Multilinear Operators on C(K, X) Spaces

Given Banach spaces X, Yand a compact Hausdorff space K, we use polymeasures to give necessary conditions for a multilinear operator from C(K, X) into Yto be completely continuous (resp. unconditionally converging). We deduce necessary and sufficient conditions for Xto have the Schur property (resp. to contain no copy of c₀), and for Kto be scattered. This extends results concerning linear operators.