Operators on 𝐶(𝐾) spaces preserving copies of Schreier spaces

Gasparis, Ioannis

doi:10.1090/S0002-9947-04-03688-8

Operators on $C(K)$ spaces preserving copies of Schreier spaces
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Trans. Amer. Math. Soc. 357 (2005), 1-30 Request permission

Abstract:

It is proved that an operator $T \colon C(K) \to X$, $K$ compact metrizable, $X$ a separable Banach space, for which the $\epsilon$-Szlenk index of $T^*(B_{X^*})$ is greater than or equal to $\omega ^\xi$, $\xi < \omega _1$, is an isomorphism on a subspace of $C(K)$ isomorphic to $X_\xi$, the Schreier space of order $\xi$. As a corollary, one obtains that a complemented subspace of $C(K)$ with Szlenk index equal to $\omega ^{\xi + 1}$ contains a subspace isomorphic to $X_\xi$.

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