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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 ${\ell }^{1}$ and $F$ has the ( RDPP), then $E{\otimes }_{\pi }F$ has the same property, provided that $L\left(E,{F}^{*}\right)=K\left(E,{F}^{*}\right)$. 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 $L\left(E,{F}^{*}\right)=K\left(E,{F}^{*}\right)$.”

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:
 46M05 Tensor products of topological linear spaces 46B25 Classical Banach spaces in the general theory of normed spaces