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Zbl 0752.46042
Emmanuele, G.
On the reciprocal Dunford-Pettis property in projective tensor products.
(English)
[J] Math. Proc. Camb. Philos. Soc. 109, No.1, 161-166 (1991). ISSN 0305-0041; ISSN 1469-8064/e

Author's abstract: We prove the following result: if a Banach space $E$ does not contain $\ell\sp 1$ and $F$ has the ( RDPP), then $E\otimes\sb \pi F$ has the same property, provided that $L(E,F\sp*)=K(E,F\sp*)$. Hence we prove that if $E\otimes\sb \pi F$ has the (RDPP) then at least one of the spaces $E$ and $F$ must not contain $\ell\sp 1$. Some corollaries are then presented as well as results concerning the necessity of the hypothesis $L(E,F\sp*)=K(E,F\sp*)$.''\par 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).
[J.B.Cooper (Linz)]
MSC 2000:
*46M05 Tensor products of topological linear spaces
46B25 Classical Banach spaces in the general theory of normed spaces

Keywords: reciprocal Dunford-Pettis property; projective tensor products; RDPP; space of compact linear operators

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