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A Classification of Tsirelson Type Spaces

Open Access article
 Printed: Oct 2008
  • J. Lopez-Abad
  • A. Manoussakis
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We give a complete classification of mixed Tsirelson spaces $T[(\mathcal F_i,\theta_i)_{i=1}^{r}]$ for finitely many pairs of given compact and hereditary families $\mathcal F_i$ of finite sets of integers and $0<\theta_i<1$ in terms of the Cantor--Bendixson indices of the families $\mathcal F_i$, and $\theta_i$ ($1\le i\le r$). We prove that there are unique countable ordinal $\alpha$ and $0<\theta<1$ such that every block sequence of $T[(\mathcal F_i,\theta_i)_{i=1}^{r}]$ has a subsequence equivalent to a subsequence of the natural basis of the $T(\mathcal S_{\omega^\alpha},\theta)$. Finally, we give a complete criterion of comparison in between two of these mixed Tsirelson spaces.
MSC Classifications: 46B20, 05D10 show english descriptions Geometry and structure of normed linear spaces
Ramsey theory [See also 05C55]
46B20 - Geometry and structure of normed linear spaces
05D10 - Ramsey theory [See also 05C55]

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