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 \(\ell^ 1\) and \(F\) has the ( RDPP), then \(E\otimes_ \pi F\) has the same property, provided that \(L(E,F^*)=K(E,F^*)\). Hence we prove that if \(E\otimes_ \pi F\) has the (RDPP) then at least one of the spaces \(E\) and \(F\) must not contain \(\ell^ 1\). 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).
Reviewer: J.B.Cooper (Linz)


46M05 Tensor products in functional analysis
46B25 Classical Banach spaces in the general theory
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