Distortion and spreading models in modified mixed Tsirelson spaces

S. A. Argyros, I. Deliyanni, and A. Manoussakis. "Distortion and spreading models in modified mixed Tsirelson spaces." Studia Mathematica 157.3 (2003): 199-236. <http://eudml.org/doc/286531>.

@article{S2003,
abstract = {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.},
author = {S. A. Argyros, I. Deliyanni, A. Manoussakis},
journal = {Studia Mathematica},
keywords = {spreading models; modified Tsirelson space; arbitrarity distortable},
language = {eng},
number = {3},
pages = {199-236},
title = {Distortion and spreading models in modified mixed Tsirelson spaces},
url = {http://eudml.org/doc/286531},
volume = {157},
year = {2003},
}

TY - JOUR
AU - S. A. Argyros
AU - I. Deliyanni
AU - A. Manoussakis
TI - Distortion and spreading models in modified mixed Tsirelson spaces
JO - Studia Mathematica
PY - 2003
VL - 157
IS - 3
SP - 199
EP - 236
AB - 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.
LA - eng
KW - spreading models; modified Tsirelson space; arbitrarity distortable
UR - http://eudml.org/doc/286531
ER -