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1
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Determining c₀ in C(𝒦) spaces

100%
Argyros S., Kanellopoulos V.
Fundamenta Mathematicae
|
2005
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tom 187
|
nr 1
61-93
EN
For a countable compact metric space 𝒦 and a seminormalized weakly null sequence (fₙ)ₙ in C(𝒦) we provide some upper bounds for the norm of the vectors in the linear span of a subsequence of (fₙ)ₙ. These bounds depend on the complexity of 𝒦 and also on the sequence (fₙ)ₙ itself. Moreover, we introduce the class of c₀-hierarchies. We prove that for every α < ω₁, every normalized weakly null sequence (fₙ)ₙ in $C(ω^{ω^{α}})$ and every c₀-hierarchy 𝓗 generated by (fₙ)ₙ, there exists β ≤ α such that a sequence of β-blocks of (fₙ)ₙ is equivalent to the usual basis of c₀.
2
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A classification of separable Rosenthal compacta and its applications

88%
Argyros S., Dodos P., Kanellopoulos V.
Instytut Matematyczny Polskiej Akademii Nauk
3
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Higher order spreading models

81%
Argyros S., Kanellopoulos V., Tyros K.
Fundamenta Mathematicae
|
2013
|
tom 221
|
nr 1
23-68
EN
We introduce higher order spreading models associated to a Banach space X. Their definition is based on ℱ-sequences $(x_{s})_{s∈ℱ}$ with ℱ a regular thin family and on plegma families. We show that the higher order spreading models of a Banach space X form an increasing transfinite hierarchy $(𝒮ℳ_{ξ}(X))_{ξ<ω₁}$. Each $𝒮ℳ_{ξ}(X)$ contains all spreading models generated by ℱ-sequences $(x_{s})_{s∈ℱ}$ with order of ℱ equal to ξ. We also study the fundamental properties of this hierarchy.
4
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Distortion and spreading models in modified mixed Tsirelson spaces

81%
Argyros S., Deliyanni I., Manoussakis A.
Studia Mathematica
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2003
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tom 157
|
nr 3
199-236
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.
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