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 ℓq⊗ˆY, the projective tensor product of ℓq and Y.