On operators which factor through $l_p$ or $c_0$

Bentuo Zheng¹

¹ Department of Mathematics Texas A&M University College Station, TX 77843, U.S.A.

Studia Mathematica, Tome 176 (2006) no. 2, pp. 177-190

Résumé

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 $c_0$.

Détail

DOI : 10.4064/sm176-2-5

Keywords: infty subspace space shrinking which satisfies block lower p estimate bounded linear operator which satisfies upper tree estimate factors through subspace sum where blocking particular prove operator infty satisfies upper tree estimate only factors through gives answer question johnson prove banach space * separable operator which satisfies upper infty estimate factors through subspace