Completely continuous multilinear mappings on L_1.

A useful result of H. Rosenthal and J. Bourgain states that, given a Banach space X, an operator T:L_1[0,1]\to X is completely continuous if and only if its composition with the natural inclusion i_\infty :L_\infty [0,1] \to L_1[0,1] is compact. We extend this result to multilinear mappings on products of L_1[0,1] spaces, and consider also the composition with the natural inclusion i:C[0,1]\to L_1[0,1]. We show that a multilinear mapping on a product of L_1[0,1] spaces is completely continuous if and only if its associated polymeasure has a relatively norm compact range. [ABSTRACT FROM AUTHOR]