On operators which factor through l p or c 0

Let 1 \u3c p \u3c ∞. Let X be a subspace of a space Z with a shrinking F.D.D. (E n) which satisfies a block lower-p estimate. Then any bounded linear operator T from X which satisfies an upper-(C, p)-tree estimate factors through a subspace of (∑ F n) lp, where (F n) is a blocking of (E n). In particular, we prove that an operator from L p (2 \u3c p \u3c ∞) satisfies an upper-(C, p)-tree estimate if and only if it factors through l p. This gives an answer to a question of W. B. Johnson. We also prove that if X is a Banach space with X* separable and T is an operator from X which satisfies an upper-(C, ∞)-estimate, then T factors through a subspace of c 0