Higher order spreading models

Argyros, Spiros A.; Kanellopoulos, Vassilis; Tyros, Konstantinos

doi:10.4064/fm221-1-2

Cited by 5 publications

(32 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover if, k > max(A l ) + 1 and if A l ∪ {k − 1} ∈ F then A l ∪ {k − 1} = A j with l < j < m + 1, and π(A j ) = π(A l ) ∪ {s} for some s has already been chosen. Thus, we need to choose π(A m+1 ) to be of the form π(A l ) ∪ {t}, where, in case that A l ∪ {k − 1} ∈ F, we need to choose t > s. Proposition 2.14 could be deduced from Corollary 2.5 and Proposition 2.6 in [3]. To make the paper as self-contained as possible we want to give a short proof.…”

Section: This Means Thatmentioning

confidence: 99%

A metric interpretation of reflexivity for Banach spaces

Motakis¹,

Schlumprecht²

2017

Duke Math. J.

View full text Add to dashboard Cite

We define two metrics d1,α and d∞,α on each Schreier family Sα, α < ω1, with which we prove the following metric characterization of reflexivity of a Banach space X: X is reflexive if and only if there is an α < ω1, so that there is no mapping Φ : Sα → X for which cd∞,α(A, B) ≤ Φ(A) − Φ(B) ≤ Cd1,α(A, B) for all A, B ∈ Sα.Secondly, we prove for separable and reflexive Banach spaces X, and certain countable ordinals α that max(Sz(X), Sz(X * )) ≤ α if and only if (Sα, d1,α) does not bi-Lipschitzly embed into X. Here Sz(Y ) denotes the Szlenk index of a Banach space Y .Contents Ostrovskii's [28, Theorem 1.2] which states that a locally finite metric space A embeds bi-Lipschitzly into a Banach space X if all of its finite subsets uniformly bi-Lipschitzly embed into X. In [18] Johnson and Schechtman characterized superflexivity, using the Diamond Graphs, D n , n ∈ N, and proved that Banach space X is super reflexive if and only if the D n , n ∈ N, do not uniformly bi-Lipschitzly embed into X. There are several other local properties, i.e., properties of the finite dimensional subspaces of Banach spaces, for which metric characterizations were found. The following are some examples: Bourgain, Milman and Wolfson [12] characterized having non trivial type using Hammond cubes (the sets B n , together with the ℓ 1 -norm), and Mendel and Naor [21,22] present metric characterizations of Banach spaces with type p, 1 < p ≤ 2, and cotype q, 2 ≤ q < ∞. For a more extensive account on the Ribe program we would like refer the reader to the survey articles [6,23] and the book [29].Instead of only asking for metric characterizations of local properties, one can also ask for metric characterizations of other properties of Banach space, properties which might not be determined by the finite dimensional subspaces. A result in this direction was obtained by Baudier, Kalton and Lancien in [9]. They showed that a reflexive Banach space X has a renorming, which is asymptoticaly uniformly convex (AUC) and asymptoticaly uniformly smooth (AUS), if and only if the countably branching trees of length n ∈ N, T n do not uniformly bi-Lipschitzly into X. Here T n = n k=0 N k , together with the graph metric, i.e., d(a, b) = i+j −max{t ≥ 0 : a s = b s , s = 1, 2 . . . t}, for a = (a 1 , a 2 , . . . a i ) = b = (b 1 , b 2 , . . . b j ) in T n . Among the many equivalent conditions for a reflexive Banach space X to be (AUC)and (AUS)-renormable (see [25]) one of them states that Sz(X) = Sz(X * ) = ω, where Sz(Z) denotes the Szlenk index of a Banach space Z (see Section 5 for the definition and properties of the Szlenk index). In [13] Dilworth, Kutzarova, Lancien and Randrianarivony, showed that a separable Banach space X is reflexive and (AUC)-and (AUS)-renormable if and only X admits an equivalent norm for which X has Rolewicz's β-property. According to [20] a Corollary 1.2. Assume that ω < α < ω 1 is an ordinal for which ω α = α. Then the following statements are equivalent for a separable and reflexive space X a) max(Sz(X), Sz(X * )) ≤ α, b) (S α , d...

show abstract

Section: This Means Thatmentioning

confidence: 99%

A metric interpretation of reflexivity for Banach spaces

Motakis¹,

Schlumprecht²

2017

Duke Math. J.

View full text Add to dashboard Cite

show abstract

“…Finally, a family F of finite subsets of N is called regular thin, if it consists of the maximal elements, under inclusion, of some regular family R. A family H of finite subsets of N is called thin if there are no s and t in H such that s is proper initial segment of t. Clearly every regular thin family is also thin. A brief presentation of the regular and the regular thin families as well as the relation between them can be found in Section 2 of [AKT2]. Finally, given a regular thin family F , an F -sequence in a Banach space is a sequence of the form (x s ) s∈F indexed by F , while an F -subsequence is a sequence of the form (x s ) s∈F ↾L indexed by F ↾ L, where L is an infinite subset of N and the restriction F ↾ L of F on L is defined by…”

Section: Introductionmentioning

confidence: 99%

“…The properties of the plegma families are explored in Section 3 of [AKT2]. Moreover, for every regular thin family F , every infinite subset L of N and every positive integer k we set…”

Section: Introductionmentioning

confidence: 99%

“…The existence of the F -spreading models is established by Theorem 4.5 in [AKT2]. For every regular family R its order o(R) is defined to be the rank of the partially ordered set R endowed with the reverse inclusion.…”

Section: Introductionmentioning

confidence: 99%

“…For every regular family R its order o(R) is defined to be the rank of the partially ordered set R endowed with the reverse inclusion. By the compactness of R its order is well defined and it is a countable ordinal number (see also Section 2 of [AKT2]). The order o(F ) of a regular thin family F is defined to be the order of the regular family for which F is the set of maximal elements under inclusion.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the structure of the set of higher order spreading models

Sarı

Tyros

2014

Studia Math.

View full text Add to dashboard Cite

We generalize some results concerning the classical notion of a spreading model for the spreading models of order ξ. Among them, we prove that the set SM w ξ (X) of the ξ-order spreading models of a Banach space X generated by subordinated weakly null F -sequences endowed with the prepartial order of domination is a semi-lattice. Moreover, if SM w ξ (X) contains an increasing sequence of length ω then it contains an increasing sequence of length ω 1 . Finally, if SM w ξ (X) is uncountable, then it contains an antichain of size the continuum.

show abstract

The stabilized set of p's in Krivine's theorem can be disconnected

Beanland

Freeman

Motakis

2015

Advances in Mathematics

View full text Add to dashboard Cite

Abstract. For any closed subset F of [1, ∞] which is either finite or consists of the elements of an increasing sequence and its limit, a reflexive Banach space X with a 1-unconditional basis is constructed so that in each block subspace Y of X, ℓp is finitely block represented in Y if and only if p ∈ F . In particular, this solves the question as to whether the stabilized Krivine set for a Banach space had to be connected. We also prove that for every infinite dimensional subspace Y of X there is a dense subset G of F such that the spreading models admitted by Y are exactly the ℓp for p ∈ G.

show abstract

Higher order spreading models

Cited by 5 publications

References 17 publications

A metric interpretation of reflexivity for Banach spaces

A metric interpretation of reflexivity for Banach spaces

On the structure of the set of higher order spreading models

The stabilized set of p's in Krivine's theorem can be disconnected

Contact Info

Product

Resources

About