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The Bourgain ell ^1-index of mixed Tsirelson space

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 نشر من قبل Wee-Kee Tang
 تاريخ النشر 2001
  مجال البحث
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Suppose that (F_n)_{n=0}^{infty} is a sequence of regular families of finite subsets of N such that F_0 contains all singletons, and (theta _n)_{n=1}^{infty} is a nonincreasing null sequence in (0,1). In this paper, we compute the Bourgain ell^1 - index of the mixed Tsirelson space T(F_0,(theta_n, F_n)_{n=1}^{infty}). As a consequence, it is shown that if eta is a countable ordinal not of the form omega^xi for some limit ordinal xi, then there is a Banach space whose ell^1-index is omega^eta . This answers a question of Judd and Odell.



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Suppose that (F_n)_{n=1}^{infty} is a sequence of regular families of finite subsets of N and (theta_n)_{n=1}^{infty} is a nonincreasing null sequence in (0,1). The mixed Tsirelson space T[(theta_{n}, F_n)_{n=1}^{infty}] is the completion of $c_{00}$ with respect to the implicitly defined norm ||x|| = max{||x||_{c_0}, sup_n sup theta_n sum_{i=1}^{j}||E_{i}x||}, where the last supremum is taken over all finite subsets E_{1},...,E_{j} of N such that E_1 < >... <E_j and {min E_1,...,min E_j} in F_n. Necessary and sufficient conditions are obtained for the existence of higher order ell ^1-spreading models in every subspace generated by a subsequence of the unit vector basis of T[(theta_{n}, F_n)_{n=1}^{infty}.
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