Illinois Journal of Mathematics

Concerning $q$-summable Szlenk index

Ryan M. Causey

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For each ordinal $\xi$ and each $1\leqslant q<\infty$, we define the notion of $\xi$-$q$-summable Szlenk index. When $\xi=0$ and $q=1$, this recovers the usual notion of summable Szlenk index. We define for an arbitrary weak$^{*}$-compact set a transfinite, asymptotic analogue $\alpha_{\xi,p}$ of the martingale type norm of an operator. We prove that this quantity is determined by norming sets and determines $\xi$-Szlenk power type and $\xi$-$q$-summability of Szlenk index. This fact allows us to prove that the behavior of operators under the $\alpha_{\xi,p}$ seminorms passes in the strongest way to injective tensor products of Banach spaces. Furthermore, we combine this fact with a result of Schlumprecht to prove that a separable Banach space with good behavior with respect to the $\alpha_{\xi,p}$ seminorm can be embedded into a Banach space with a shrinking basis and the same behavior under $\alpha_{\xi,p}$, and in particular it can be embedded into a Banach space with a shrinking basis and the same $\xi$-Szlenk power type. Finally, we completely elucidate the behavior of the $\alpha_{\xi,p}$ seminorms under $\ell_{r}$ direct sums. This allows us to give an alternative proof of a result of Brooker regarding Szlenk indices of $\ell_{p}$ and $c_{0}$ direct sums of operators.

Article information

Illinois J. Math., Volume 62, Number 1-4 (2018), 381-426.

Received: 27 November 2018
Revised: 27 November 2018
First available in Project Euclid: 13 March 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46B03: Isomorphic theory (including renorming) of Banach spaces 46B06: Asymptotic theory of Banach spaces [See also 52A23]
Secondary: 46B28: Spaces of operators; tensor products; approximation properties [See also 46A32, 46M05, 47L05, 47L20] 47B10: Operators belonging to operator ideals (nuclear, p-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20]


Causey, Ryan M. Concerning $q$-summable Szlenk index. Illinois J. Math. 62 (2018), no. 1-4, 381--426. doi:10.1215/ijm/1552442668.

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