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Keywords:
completely continuous; unconditionally converging; multilinear operators; $C(K,X)$ spaces
Summary:
Given Banach spaces~ $X$, $Y$ and a compact Hausdorff space~ $K$, we use polymeasures to give necessary conditions for a multilinear operator from $C(K,X)$ into~ $Y$ to be completely continuous (resp.~ unconditionally converging). We deduce necessary and sufficient conditions for~ $X$ to have the Schur property (resp.~ to contain no copy of~ $c_0$), and for~ $K$ to be scattered. This extends results concerning linear operators.
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