Mixed Tsirelson Research Articles

A Study of Tsirelson and Schreier Spaces

This paper is a short review of some results in Tsirelson space. The concept of the paper is to include some well-known theorems on Tsirelson’s space constructed by Figiel and Johnson together with results concerning holomorphic functions on the classical and modified mixed Tsirelson spaces.

Positional graphs and conditional structure of weakly null sequences

We prove that, unless assuming additional set theoretical axioms, there are no reflexive spaces without unconditional sequences of the density continuum. We show that for every integer n there are normalized weakly-null sequences of length ωn without unconditional subsequences. This together with a result of Dodos et al. (2011) [7] shows that ωω is the minimal cardinal κ that could possibly have the property that every weakly null κ-sequence has an infinite unconditional basic subsequence. We also prove that for every cardinal number κ which is smaller than the first ω-Erdős cardinal there is a normalized weakly-null sequence without subsymmetric subsequences. Finally, we prove that mixed Tsirelson spaces of uncountable densities must always contain isomorphic copies of either c0 or ℓp, with p≥1.

Strictly singular non-compact operators on a class of HI spaces

We present a method for constructing bounded strictly singular non-compact operators on mixed Tsirelson spaces defined either by the families (A_n) or (S_n) of a certain class, as well as on spaces built on them, including hereditarily indecomposable spaces.

Isomorphisms and strictly singular operators in mixed Tsirelson spaces

We study the family of isomorphisms and strictly singular operators in mixed Tsirelson spaces and their modified versions setting. We show sequential minimality of modified mixed Tsirelson spaces TM[(Sn,θn)n] satisfying some regularity conditions and present results on existence of strictly singular non-compact operators on subspaces of mixed Tsirelson spaces defined by the families (An)n and (Sn)n.

Quasiminimality in mixed Tsirelson spaces

AbstractWe prove quasiminimality of regular the mixed Tsirelson spaces \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$T[(\mathcal {S}_{n},\theta _{n})_{n}]$\end{document} with the sequence \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\big (\frac{\theta _n}{\theta ^n}\big )_n$\end{document} decreasing, where θ = limnθ1/nn, and quasiminimality of all mixed Tsirelson spaces \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$T[(\mathcal {A}_{n},\theta _{n})_{n}]$\end{document}. We prove that under certain assumptions on the sequence (θn)n the dual spaces are quasiminimal.

A Classification of Tsirelson Type Spaces

AbstractWe give a complete classification of mixed Tsirelson spacesfor finitely many pairs of given compact and hereditary familiesof finite sets of integers and 0 <θi< 1 in terms of the Cantor–Bendixson indices of the families, andθi(1 ≤i≤r). We prove that there are unique countable ordinalαand 0 <θ< 1 such that every block sequence ofhas a subsequence equivalent to a subsequence of the natural basis of the. Finally, we give a complete criterion of comparison in between two of these mixed Tsirelson spaces.

Minimality properties of Tsirelson type spaces

We study minimality properties of partly modified mixed Tsirelson spaces. A Banach space with a normalized basis $(e_{k})$ is said to be <i>subsequentially minimal</i> if for every normalized block basis $(x_{k})$ of $(e_{k}) ,$ there is a further block b

More mixed Tsirelson spaces that are not isomorphic to their modified versions

The class of mixed Tsirelson spaces is an important source of examples in the recent development of the structure theory of Banach spaces. The related class of modified mixed Tsirelson spaces has also been well studied. In the present paper, we investigate the problem of comparing isomorphically the mixed Tsirelson space $T[({\mathcal{S}}_{n},\theta_{n})_{n=1}^{\infty}]$ and its modified version $T_{M}[({\mathcal{S}}_{n},\theta_{n})_{n=1}^{\infty }]$. It is shown that these spaces are not isomorphic for a large class of parameters $(θ_n)$.

L1-Spreading models in subspaces of mixed Tsirelson spaces

We investigate the existence of higher order $\ell^{1}$-spreading models in subspaces of mixed Tsirelson spaces. For instance, we show that the following conditions are equivalent for the mixed Tsirelson space $X = T[(\theta _{n},{\mathcal{S}}_{n})^{\inft

On Total Incomparability of Mixed Tsirelson Spaces

We give criteria of total incomparability for certain classes of mixed Tsirelson spaces. We show that spaces of the form $$T\left[ {\left( {\mathcal{M}_k ,\theta _k } \right)_{k = 1}^l } \right]$$ with index $$i\left( {\mathcal{M}_k } \right)$$ finite are either c 0 or $$\ell _p $$ saturated for some p and we characterize when any two spaces of such a form are totally incomparable in terms of the index $$i\left( {\mathcal{M}_k } \right)$$ and the parameter θ k . Also, we give sufficient conditions of total incomparability for a particular class of spaces of the form $$T\left[ {\left( {\mathcal{A}_k ,\theta _k } \right)_{k = 1}^\infty } \right]$$ in terms of the asymptotic behaviour of the sequence $$\left\| {\sum\limits_{j = 1}^n {e_i } } \right\|$$ where (e i is the canonical basis.

Distortion and spreading models in modified mixed Tsirelson spaces

The results of the first part concern the existence of higher order $\ell_{1}$ spreading models in asymptotic $\ell_{1}$ Banach spaces. We sketch the proof of the fact that the mixed Tsirelson space $T[({\cal S}_{n},\theta_{n})_{n}]$, $\theta_{n+m}\geq\th

Some remarks on spreading models and mixed Tsirelson spaces

We prove that if a Banach space with a bimonotone shrinking basis does not contain l w 1 spreading models but every block sequence of the basis contains a further block sequence which is a c - l n 1 spreading model for every n E N, then every subspace has a further subspace which is arbitrarily distortable. We also prove that a mixed Tsirelson space T[(S n ,θ n ) n ], such that θ n ? 0, does not contain l w2 1 spreading models.

Interpolating hereditarily indecomposable Banach spaces

A Banach space X is said to be Hereditarily Indecomposable (H.I.) if for any pair of closed subspaces Y , Z of X with Y ∩ Z = {0}, Y + Z is not a closed subspace. (Throughout this section by the term “subspace” we mean a closed infinite-dimensional subspace of X .) The H.I. spaces form a new and, as we believe, fundamental class of Banach spaces. The celebrated example of a Banach space with no unconditional basic sequence, due to W. Gowers and B. Maurey ([GM]), is the first construction of a H.I. space. It is easily seen that every H.I. space does not contain any unconditional basic sequence. Actually, the concept of H.I. spaces came after W. Johnson’s observation that this was a property of the Gowers-Maurey example. To describe even further the peculiar structure of a H.I. space, we recall an alternative definition of such a space. A Banach space X is a H.I. space if and only if for every pair of subspaces Y , Z and e > 0 there exist y ∈ Y , z ∈ Z with ||y|| = ||z|| = 1 and ||y − z|| < e. Thus, H.I. spaces are structurally irrelevant to classical Banach spaces, in particular to Hilbert spaces. Other constructions of H.I. spaces already exist. We mention Argyros and Deliyanni’s construction of H.I. spaces which are asymptotic ` spaces ([AD2]), V. Ferenczi’s example of a uniformly convex H.I. space ([F2]) and H.I. modified asymptotic ` spaces contained in [ADKM]. Other examples of Banach spaces which are H.I. or which have a H.I. subspace are given in [G1], [H], [OS1]. The construction of such a space requires several steps and it uses two fundamental ideas. The first is Tsirelson’s recursive definition of saturated norms ([Ts]) and the second is Maurey-Rosenthal’s construction of weakly null sequences without unconditional basic subsequences ([MR]). An important ingredient in the GowersMaurey construction is the Schlumprecht space. This is a Tsirelson type Banach space which is arbitrarily distortable and has been used in the solution of important problems. Thus beyond its use in the constructions of H.I. spaces it plays a central role in the solution of the distortion problem for Hilbert spaces ([OS]). The essential difference between Schlumprecht and Tsirelson spaces became more transparent in [AD2] where the mixed Tsirelson spaces were introduced. It is natural to expect that H.I. spaces share special and interesting properties not found in the previously known Banach spaces. Indeed, the following theorem is proven in

Distorting mixed tsirelson spaces

Any regular mixed Tsirelson spaceT(θ n ,S n )N for whichθ n /θ n → 0, whereθ=lim n θ 1 , is shown to be arbitrarily distortable. Certain asymptoticl 1 constants for those and other mixed Tsirelson spaces are calculated. Also, a combinatorial result on the Schreier families (S α ) α<ω 1 is proved and an application is given to show that for every Banach spaceX with a basis (e i ), the two Δ-spectrums Δ(X) and Δ(X, (e i )) coincide.

Modified Mixed Tsirelson Spaces

We study the modified and boundedly modified mixed Tsirelson spacesTM[(Fkn,θn)∞n=1] andTM(s)[(Fkn,θn)∞n=1], respectively, defined by a subsequence (Fkn) of the sequence of Schreier families (Fn). These are reflexive asymptotic ℓ1spaces with an unconditional basis (ei)ihaving the property that every sequence {xi}ni=1of normalized disjointly supported vectors contained in ⦠ei⦔∞i=nis equivalent to the basis of ℓn1. We show that if limθ1/nn=1 then the spaceT[(Fn,θn)∞n=1] and its modified variationsTM[(Fn,θn)∞n=1] orTM(s)[(Fn,θn)∞n=1] are totally incomparable by proving thatc0is finitely disjointly representable in every block subspace ofT[(Fn,θn)∞n=1]. Next, we present an example of a boundedly modified mixed Tsirelson spaceXM(1),u=TM(1)[(Fkn,θn)∞n=1] which is arbitrarily distortable. Finally, we construct a variation of the spaceXM(1),uwhich is hereditarily indecomposable.