Distortion and spreading models in modified mixed Tsirelson spaces

S. A. Argyros ; I. Deliyanni ; A. Manoussakis

Studia Mathematica, Tome 157 (2003), p. 199-236 / Harvested from The Polish Digital Mathematics Library

Résumé

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[(ₙ,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 ₙ-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 $T_{M}[(_{kₙ},θₙ)ₙ]$ is arbitrarily distortable. Moreover, for an appropriate choice of (kₙ,θₙ)ₙ, every block subspace admits an $ℓ₁^{ω}$ spreading model.

Détail

Publié le : 2003-01-01

Zbl 1028.46019

EUDML-ID : urn:eudml:doc:286531