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تسجيل مستخدم جديدell^1-spreading models in subspaces of mixed Tsirelson spaces
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We investigate the existence of higher order ell^1-spreading models in subspaces of mixed Tsirelson spaces. For instance, we show that the following conditions are equivalent for the mixed Tsirelson space X=T[(theta _n,S_n)_{n=1}^{infty}] (1)Every block subspace of $X$ contains an ell^1-S_{omega}-spreading model, (2)The Bourgain ell^1-index I_b(Y) = I(Y) > omega^{omega} for any block subspace Y of X, (3)lim_mlimsup_ntheta_{m+n}/theta_n > 0 and every block subspace Y of X contains a block sequence equivalent to a subsequence of the unit vector basis of X. Moreover, if one (and hence all) of these conditions holds, then X is arbitrarily distortable.
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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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aken over all finite subsets E_{1},...,E_{j} of $mathbb{N}$ such that $E_{1}<...<E_{j}$ and {min E_{1},...,min E_{j}} in S_beta. It is shown that the Bourgain $ell^{1}$-index of tilde{T}_{alpha,beta} is omega^{alpha+beta.omega}. In particular, if alpha =omega^{alpha_{1}}. m_{1}+...+omega^{alpha_{n}}. m_{n} in Cantor normal form and alpha_{n} is not a limit ordinal, then there exists a Banach space whose ell^{1}-index is omega^{alpha}.
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