Dunford-Pettis like properties on tensor products

Abstract

In this paper we study equivalent formulations of the DP*P_p (1 < p < ∞). We show that X has the DP*P_p if and only if every weakly-p-Cauchy sequence in X is a limited subset of X. We give sufficient conditions on Banach spaces X and Y so that the projective tensor product X ⨂_πY , the dual (X  ⨂_ϵ Y )* of their injective tensor product, and the bidual (X ⨂_πY ) ** of their projective tensor product, do not have the DPP_p, 1 < p < ∞. We also show that in some cases, the projective and the injective tensor products of two spaces do not have the DP * P_p, 1 < p < ∞.

Mathematics Subject Classification (2010): 46B20, 46B25, 46B28.

Keywords: The Dunford-Pettis property of order p, the DP*-property of order p, p-convergent operator, injective and projective tensor products