Remarks on Weak Compactness of Operators Defined on Certain Injective Tensor Products

We show that if X is a L∞-space with the Dieudonnè property and Y is a Banach space not containing l1, then any operator T : X⊗ Y → Z, where Z is a weakly sequentially complete Banach space, is weakly compact. Some other results of the same kind are obtained. Let X be a L∞-space (see [1] for this notion and some useful results on L∞-spaces) and Y be a Banach space not containing l1.We consider the injective tensor productX⊗ Y (see [3]), and we investigate the following problem: when is any operator T : X ⊗ Y → Z, where Z is a Banach space, weakly compact ? In the case of X = C(K) there are some papers devoted to the study of this question (see [2, 6-9]), but nothing seems to be known in the present setting; we observe that the theorems proved in the paper extend all of the above-quoted results, but their proofs make use of the results of the results in [2, 8], so that they may be considered interesting complements to those theorems. Because the proofs of our results are similar, we give the proof of Theorem 2 only and leave the others to the reader. We need the following definition: a Banach space E has the Dieudonnè property if any weakly completely continuous (or Dieudonnè) operator defined on it is weakly compact [8]. Received by the editors March 22, 1991. 1991 Mathematics Subject Classification. Primary 47B99,46M05