## 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)

### MSC:

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

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