Distortion and spreading models in modified mixed Tsirelson spaces

• Autorzy: S. A. Argyros , I. Deliyanni , A. Manoussakis
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46B20: Geometry and structure of normed linear spaces
• 46B03: Isomorphic theory (including renorming) of Banach spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2003
• Tom: 157
• Numer: 3
• Strony: 199-236

Daty

wydano

2003

Twórcy

autor

S. A. Argyros

• Department of Mathematics, National Technical University of Athens, Athens, Greece

autor

I. Deliyanni

• Department of Mathematics, University of Crete, Heraklion, Crete, Greece

autor

A. Manoussakis

• Department of Sciences, Technical University of Crete, Chania, Crete, Greece

Identyfikatory

DOI

10.4064/sm157-3-1