On the Dunford–Pettis property of tensor product spaces

Ioana Ghenciu¹

¹ Mathematics Department University of Wisconsin-River Falls River Falls, WI 54022, U.S.A.

Colloquium Mathematicum, Tome 125 (2011) no. 2, pp. 221-231.

Résumé

We give sufficient conditions on Banach spaces $E$ and $F$ so that their projective tensor product $E\otimes _\pi F$ and the duals of their projective and injective tensor products do not have the Dunford–Pettis property. We prove that if $E^*$ does not have the Schur property, $F$ is infinite-dimensional, and every operator $T:E^*\to F^{**}$ is completely continuous, then $(E\otimes _\epsilon F)^*$ does not have the DPP. We also prove that if $E^*$ does not have the Schur property, $F$ is infinite-dimensional, and every operator $T: F^{**} \to E^*$ is completely continuous, then $(E\otimes _\pi F)^*\simeq L(E,F^*)$ does not have the DPP.

Détail

DOI : 10.4064/cm125-2-7

Keywords: sufficient conditions banach spaces their projective tensor product otimes duals their projective injective tensor products have dunford pettis property prove * does have schur property infinite dimensional every operator * ** completely continuous otimes epsilon * does have dpp prove * does have schur property infinite dimensional every operator ** * completely continuous otimes * simeq * does have dpp