Completely continuous multilinear mappings on 𝐿₁

A useful result of H. Rosenthal and J. Bourgain states that, given a Banach space X X, an operator T: L 1 0, 1 → X T: L₁0, 1 X is completely continuous if and only if its composition with the natural inclusion i ∞: L ∞ 0, 1 → L 1 0, 1 i_: L_ 0, 1 L₁0, 1 is compact. We extend this result to multilinear mappings on products of L 1 0, 1 L₁0, 1 spaces, and consider also the composition with the natural inclusion i: C 0, 1 → L 1 0, 1 i: C0, 1 L₁0, 1. We show that a multilinear mapping on a product of L 1 0, 1 L₁0, 1 spaces is completely continuous if and only if its associated polymeasure has a relatively norm compact range.