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Abstract

Let $1< p< \infty$. 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 $(\sum F_n)_{l_p}$, where $(F_n)$ is a blocking of $(E_n)$. In particular, we prove that an operator from $L_p\, (2< p< \infty)$ 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-$(C,\infty)$-estimate, then $T$ factors through a subspace of $c_0$.