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:L_1[0,1]→ 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] → L_1[0,1] is compact. We extend this result to multilinear mappings on products of L 1 [ 0 , 1 ] L_1[0,1] spaces, and consider also the composition with the natural inclusion i : C [ 0 , 1 ] → L 1 [ 0 , 1 ] i:C[0,1]→ L_1[0,1] . We show that a multilinear mapping on a product of L 1 [ 0 , 1 ] L_1[0,1] spaces is completely continuous if and only if its associated polymeasure has a relatively norm compact range.
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Cilia et al. (2024) studied this question.