A Classification of Tsirelson Type Spaces

Lopez-Abad, J. ; Manoussakis, A.

Canadian journal of mathematics, Tome 60 (2008) no. 5, pp. 1108-1148

Résumé

We give a complete classification of mixed Tsirelson spaces $T\left[ ({{F}_{i,}}{{\theta }_{i}})_{i=1}^{r} \right]$ for finitely many pairs of given compact and hereditary families ${{F}_{i}}$ of finite sets of integers and $0<{{\theta }_{i}}<1$ in terms of the Cantor–Bendixson indices of the families ${{F}_{i}}$ , 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 $T\left[ ({{F}_{i,}}{{\theta }_{i}})_{i=1}^{r} \right]$ has a subsequence equivalent to a subsequence of the natural basis of the $T({{S}_{{{\omega }^{\alpha }}}},\theta )$ . Finally, we give a complete criterion of comparison in between two of these mixed Tsirelson spaces.

Détail

DOI : 10.4153/CJM-2008-049-0

Mots-clés : 46B20, 05D10

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