Dunford-Pettis like properties on tensor products

Abstract

In this paper we study equivalent formulations of the DP∗ Pp (1 < p < ∞). We show that X has the DP∗ Pp 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 Pp, 1 < p < ∞. We also show that in some cases, the projective and the injective tensor products of two spaces do not have the DP∗ Pp, 1 < p < ∞.