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On the reciprocal Dunford-Pettis property in projective tensor products. (English) Zbl 0752.46042

Author’s abstract: “We prove the following result: if a Banach space E does not contain 1 and F has the ( RDPP), then E π F has the same property, provided that L(E,F * )=K(E,F * ). Hence we prove that if E π F has the (RDPP) then at least one of the spaces E and F must not contain 1 . Some corollaries are then presented as well as results concerning the necessity of the hypothesis L(E,F * )=K(E,F * ).”

Here K denotes the space of compact linear operators and the symbol (RDPP) denotes the property of the title which can be defined in the following intrinsic manner: E has (RDPP) if bounded subsets M of the dual with the property that each sequence in E which converges weakly to zero converges uniformly on M are realtively weakly compact. (The original definition involves Dunford-Pettis operators acting on the space — the equivalence with the above formulation is a result of Leavelle).

MSC:
46M05Tensor products of topological linear spaces
46B25Classical Banach spaces in the general theory of normed spaces