Some mapping properties of p-summing adjoint operators

Let X and Y be Banach spaces and u be a continuous linear operator from X to Y . We prove that if u∗, the adjoint operator of u, is p-summing for some p 1, then for any q 2, u takes (almost) unconditionally summable sequences in X into members of q ⊗ˆ Y , the projective tensor product of q and Y .  2004 Elsevier Inc. All rights reserved