Author’s abstract: “We prove the following result: if a Banach space does not contain and has the ( RDPP), then has the same property, provided that . Hence we prove that if has the (RDPP) then at least one of the spaces and must not contain . Some corollaries are then presented as well as results concerning the necessity of the hypothesis .”
Here 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: has (RDPP) if bounded subsets of the dual with the property that each sequence in which converges weakly to zero converges uniformly on 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).