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2003 | 157 | 3 | 199-236
Tytuł artykułu

Distortion and spreading models in modified mixed Tsirelson spaces

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The results of the first part concern the existence of higher order ℓ₁ spreading models in asymptotic ℓ₁ Banach spaces. We sketch the proof of the fact that the mixed Tsirelson space T[(𝓢ₙ,θₙ)ₙ], $θ_{n+m} ≥ θₙθₘ$ and $lim_{n} θₙ^{1/n} = 1$, admits an $ℓ₁^{ω}$ spreading model in every block subspace. We also prove that if X is a Banach space with a basis, with the property that there exists a sequence (θₙ)ₙ ⊂ (0,1) with $lim_{n} θₙ^{1/n} = 1$, such that, for every n ∈ ℕ, $||∑_{k=1}^{m} x_{k}|| ≥ θₙ∑_{k=1}^{m} ||x_{k}||$ for every 𝓢ₙ-admissible block sequence $(x_{k})_{k=1}^{m}$ of vectors in X, then there exists c > 0 such that every block subspace of X admits, for every n, an ℓ₁ⁿ spreading model with constant c. Finally, we give an example of a Banach space which has the above property but fails to admit an $ℓ₁^{ω}$ spreading model.
In the second part we prove that under certain conditions on the double sequence (kₙ,θₙ)ₙ the modified mixed Tsirelson space $T_{M}[(𝓢_{kₙ},θₙ)ₙ]$ is arbitrarily distortable. Moreover, for an appropriate choice of (kₙ,θₙ)ₙ, every block subspace admits an $ℓ₁^{ω}$ spreading model.
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  • Department of Mathematics, National Technical University of Athens, Athens, Greece
autor
  • Department of Mathematics, University of Crete, Heraklion, Crete, Greece
  • Department of Sciences, Technical University of Crete, Chania, Crete, Greece
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm157-3-1
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