On operators which factor through l p or c 0
Let 1 < p < ∞. 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 < p < ∞) 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.
Zheng, B. (2006). On operators which factor through l p or c 0. Studia Mathematica, 176 (2), 177-190. https://doi.org/10.4064/sm176-2-5