On operators from separable reflexive spaces with asymptotic structure
Volume 185 / 2008
Abstract
Let $1< q< p< \infty$ and $q\leq r\leq p$. Let $X$ be a reflexive Banach space satisfying a lower-$\ell_q$-tree estimate and let $T$ be a bounded linear operator from $X$ which satisfies an upper-$\ell_p$-tree estimate. Then $T$ factors through a subspace of $(\sum F_n)_{\ell_r}$, where $(F_n)$ is a sequence of finite-dimensional spaces. In particular, $T$ factors through a subspace of a reflexive space with an $(\ell_p, \ell_q)$ FDD. Similarly, let $1< q< r< p< \infty$ and let $X$ be a separable reflexive Banach space satisfying an asymptotic lower-$\ell_q$-tree estimate. Let $T$ be a bounded linear operator from $X$ which satisfies an asymptotic upper-$\ell_p$-tree estimate. Then $T$ factors through a subspace of $(\sum G_n)_{\ell_r}$, where $(G_n)$ is a sequence of finite-dimensional spaces. In particular, $T$ factors through a subspace of a reflexive space with an asymptotic $(\ell_p, \ell_q)$ FDD.
