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\(\ell^1\)-spreading models in mixed Tsirelson space. (English) Zbl 1081.46007

The authors study the class of Banach spaces called mixed Tsirelson space \(T[(\theta_n,\mathcal F_n)_{n=1}^\infty]\) where each \(\mathcal F_n\) is a regular family of finite subsets of \(\mathbb N\). These were first described by S. A. Argyros and I. Deliyanni [Trans. Am. Math. Soc. 349, No. 3, 973–995 (1997; Zbl 0869.46002)] and generalize Tsirelson’s space \(T\).
In [Proc. Am. Math. Soc. 131, No. 2, 511–521 (2003; Zbl 1033.46005)], the authors studied the Bourgain \(\ell_1\) ordinal index of such spaces. In the present paper, they are concerned with higher order \(\ell_1\) spreading models contained in any subspace spanned by a subsequence of the unit vector basis. A main tool is I. Gasparis’ dichotomy theorem [Proc. Am. Math. Soc. 129, No. 3, 759–764 (2001; Zbl 0962.46006)].

MSC:

46B03 Isomorphic theory (including renorming) of Banach spaces
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B20 Geometry and structure of normed linear spaces
46B45 Banach sequence spaces
03E05 Other combinatorial set theory
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References:

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[2] Argyros, S. A.; Deliyanni, I., Examples of asymptotic ℓ^1 Banach spaces, Transactions of the American Mathematical Society, 349, 973-995 (1997) · Zbl 0869.46002 · doi:10.1090/S0002-9947-97-01774-1
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[8] Leung, D.; Tang, W.-K., The ℓ^1-indices of Tsirelson type spaces, Proceedings of the American Mathematical Society, 131, 511-521 (2003) · Zbl 1033.46005 · doi:10.1090/S0002-9939-02-06586-3
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