On operators from separable reflexive spaces with asymptotic structure

Bentuo Zheng¹

¹ Department of Mathematics The University of Texas at Austin 1 University Station C1200 Austin, TX 78712-0257, U.S.A.

Studia Mathematica, Tome 185 (2008) no. 1, pp. 87-98

Résumé

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.

Détail

DOI : 10.4064/sm185-1-6

Keywords: infty leq leq reflexive banach space satisfying lower ell q tree estimate bounded linear operator which satisfies upper ell p tree estimate factors through subspace sum ell where sequence finite dimensional spaces particular factors through subspace reflexive space ell ell nbsp fdd similarly infty separable reflexive banach space satisfying asymptotic lower ell q tree estimate bounded linear operator which satisfies asymptotic upper ell p tree estimate factors through subspace sum ell where sequence finite dimensional spaces particular factors through subspace reflexive space asymptotic ell ell nbsp fdd