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 ⊗_π Y, the dual (X ⊗_ϵ Y)^∗ of their injective tensor product, and the bidual (X ⊗_π Y)^∗∗ of their projective tensor product, do not have the DP P_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 < ∞.