arXiv subjects

Spiros A. Argyros

Publications and source records attributed to Spiros A. Argyros.

At least 19 recordsLinked to original sources

arXiv2026

Explicity Defined Norms on $JT_*$ Spaces

We present a Banach space $X_α$, equipped with an explicit norm, whose properties resemble those of the predual $JT_*$ of the James Tree space. In particular, $X_α$ admits a shrinking dyadic basis with the property that the sums of its elements over any segment are uniformly bounded. The space $X_α$ is complementably $\ell_2$-saturated, and its dual $X_α^*$ does not contain an isomorphic copy of $\ell_1$. Finally, the bidual $X_α^{**}$ is also complementably $\ell_2$-saturated, and the quotient $X_α^{**}/X_α$ is isometric to $c_0(2^{\mathbb{N}})$.

math.FA↗

arXiv2026

A Study of the Extreme Points in the Unit Ball of $JT$

In this note, we investigate the extreme points of the unit ball of the James Tree space ($JT$). We relate the geometric structure of $JT$ to the classical James space $J$ and provide partial characterizations of extremality based on the concept of separated vectors. We provide a complete characterization for positive vectors and establish the equal sums property for positive extreme points.

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arXiv2026

Embedding $\ell_2$ and $J$ into subspaces of $JT$ and $JT^*$

In the first part of the paper we show that every closed subspace of $JT$ or $JT^*$ contains $\ell_2$ complemented in $JT$ or $JT^*$ respectively, and $JT$ contains uncomplemented copies of $\ell_2$. As a result, the predual $\B$ of $JT$, as well as the spaces $JT$ and $JT^*$, are subprojective and superprojective. In the second part, we prove that every weakly Cauchy sequence that is not weakly convergent in $JT$ has a subsequence equivalent to the basis of $J$. Hence, every non-reflexive subspace of $JT$ contains an isomorphic copy of $J$, and every Schauder basic sequence in $JT$ has a subsequence which is equivalent either to the basis of $\ell_2$ or to the basis of $J$. Moreover these subspaces may be selected to be complemented in $JT$.

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arXiv2025

The Extreme Points of the Unit Ball of the James space $J$ and its dual spaces

We provide a new proof of S. Bellenot's characterization of the extreme points of the unit ball $B_J$ of James quasi-reflexive space $J$. We also provide an explicit description of the norm of $J^{**}$ which yields an analogous characterization for the extreme points of $B_{J^{**}}$. In the last part of the paper we describe the set of all extreme points of $B_{J^*}$ and its norm closure. It is remarkable that the descriptions of the extreme points of $B_J$ and $B_{J^*}$ are closely connected.

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arXiv2024

The HI extension of the standard HI spaces

A Hereditarily Indecomposable (HI) Banach space $X$ admits an HI extension if there exists an HI space $Z$ such that $X$ is isomorphic to a subspace $Y$ of $Z$ and $Z/Y$ is of infinite dimension. The problem whether or not every HI space admits an HI extension is attributed to A. Pelczynski. In this paper we present a method to define HI-extensions of the standard HI spaces, a class which includes the Gowers-Maurey space, asymptotic $\ell_{p}$-HI spaces and others.

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arXiv2023

The complete separation of the two finer asymptotic $\ell_{p}$ structures for $1\le p<\infty$

For $1\le p <\infty$, we present a reflexive Banach space $\mathfrak{X}^{(p)}_{\text{awi}}$, with an unconditional basis, that admits $\ell_p$ as a unique asymptotic model and does not contain any Asymptotic $\ell_p$ subspaces. D. Freeman, E. Odell, B. Sari and B. Zheng have shown that whenever a Banach space not containing $\ell_1$, in particular a reflexive Banach space, admits $c_0$ as a unique asymptotic model then it is Asymptotic $c_0$. These results provide a complete answer to a problem posed by L. Halbeisen and E. Odell and also complete a line of inquiry of the relation between specific asymptotic structures in Banach spaces, initiated in a previous paper by the first and fourth authors. For the definition of $\mathfrak{X}^{(p)}_{\text{awi}}$ we use saturation with asymptotically weakly incomparable constraints, a new method for defining a norm that remains small on a well-founded tree of vectors which penetrates any infinite dimensional closed subspace.

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arXiv2020

Non-asymptotic $\ell_1$ spaces with unique $\ell_1$ asymptotic model

A recent result of Freeman, Odell, Sari, and Zheng states that whenever a separable Banach space not containing $\ell_1$ has the property that all asymptotic models generated by weakly null sequences are equivalent to the unit vector basis of $c_0$ then the space is Asymptotic $c_0$. We show that if we replace $c_0$ with $\ell_1$ then this result is no longer true. Moreover, a stronger result of B. Maurey - H. P. Rosenthal type is presented, namely, there exists a reflexive Banach space with an unconditional basis admitting $\ell_1$ as a unique asymptotic model whereas any subsequence of the basis generates a non-Asymptotic $\ell_1$ subspace.

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arXiv2019

On the complete separation of asymptotic structures in Banach spaces

Let $(e_i)_i$ denote the unit vector basis of $\ell_p$, $1\leq p< \infty$, or $c_0$. We construct a reflexive Banach space with an unconditional basis that admits $(e_i)_i$ as a uniformly unique spreading model while it has no subspace with a unique asymptotic model, and hence it has no asymptotic-$\ell_p$ or $c_0$ subspace. This solves a problem of E. Odell. We also construct a space with a unique $\ell_1$ spreading model and no subspace with a uniformly unique $\ell_1$ spreading model. These results are achieved with the utilization of a new version of the method of saturation under constraints that uses sequences of functionals with increasing weights.

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arXiv2016

A study of conditional spreading sequences

It is shown that every conditional spreading sequence can be decomposed into two well behaved parts, one being unconditional and the other being convex block homogeneous, i.e. equivalent to its convex block sequences. This decomposition is then used to prove several results concerning the structure of spaces with conditional spreading bases as well as results in the theory of conditional spreading models. Among other things, it is shown that the space $C(ω^ω)$ is universal for all spreading models, i.e., it admits all spreading sequences, both conditional and unconditional, as spreading models. Moreover, every conditional spreading sequence is generated as a spreading model by a sequence in a space that is quasi-reflexive of order one.

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arXiv2016

An Indecomposable and unconditionally saturated Banach space

We construct an indecomposable reflexive Banach space $X_{ius}$ such that every infinite dimensional closed subspace contains an unconditional basic sequence. We also show that every operator $T\in \mathcal{B}(X_{ius})$ is of the form $λI+S$ with $S$ a strictly singular operator.

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arXiv2016

The scalar-plus-compact property in spaces without reflexive subspaces

A hereditarily indecomposable Banach space $\mathfrak{X}_{\mathfrak{nr}}$ is constructed that is the first known example of a $\mathscr{L}_\infty$-space not containing $c_0$, $\ell_1$, or reflexive subspaces and answers a question posed by J. Bourgain. Moreover, the space $\mathfrak{X}_{\mathfrak{nr}}$ satisfies the "scalar-plus-compact" property and it is the first known space without reflexive subspaces having this property. It is constructed using the Bourgain-Delbaen method in combination with a recent version of saturation under constraints in a mixed-Tsirelson setting. As a result, the space $\mathfrak{X}_{\mathfrak{nr}}$ has a shrinking finite dimensional decomposition and does not contain a boundedly complete sequence.

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arXiv2015

On the structure of separable $\mathcal{L}_\infty$-spaces

Based on a construction method introduced by J. Bourgain and F. Delbaen, we give a general definition of a Bourgain-Delbaen space and prove that every infinite dimensional separable $\mathcal{L}_\infty$-space is isomorphic to such a space. Furthermore, we provide an example of a $\mathcal{L}_\infty$ and asymptotic $c_0$ space not containing $c_0$.

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arXiv2015

A dual method of constructing hereditarily indecomposable Banach spaces

A new method of defining hereditarily indecomposable Banach spaces is presented. This method provides a unified approach for constructing reflexive HI spaces and also HI spaces with no reflexive subspace. All the spaces presented here satisfy the property that the composition of any two strictly singular operators is a compact one. This yields the first known example of a Banach space with no reflexive subspace such that every operator has a non-trivial closed invariant subspace.

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arXiv2014

A hereditarily indecomposable Banach space with rich spreading model structure

We present a reflexive Banach space $\mathfrak{X}_{_{^\text{usm}}}$ which is Hereditarily Indecomposable and satisfies the following properties. In every subspace $Y$ of $\mathfrak{X}_{_{^\text{usm}}}$ there exists a weakly null normalized sequence $\{y_n\}_n$, such that every subsymmetric sequence $\{z_n\}_n$ is isomorphically generated as a spreading model of a subsequence of $\{y_n\}_n$. Also, in every block subspace $Y$ of $\mathfrak{X}_{_{^\text{usm}}}$ there exists a seminormalized block sequence $\{z_n\}$ and $T:\mathfrak{X}_{_{^\text{usm}}}\rightarrow\mathfrak{X}_{_{^\text{usm}}}$ an isomorphism such that for every $n\in\mathbb{N}$ $T(z_{2n-1}) = z_{2n}$. Thus the space is an example of an HI space which is not tight by range in a strong sense.

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arXiv2014

$α$-Large Families and Applications to Banach Space Theory

The notion of $α$-large families of finite subsets of an infinite set is defined for every countable ordinal number $α$, extending the known notion of large families. The definition of the $α$-large families is based on the transfinite hierarchy of the Schreier families $\mathcal{S}_α, α<ω_1$. We prove the existence of such families on the cardinal number $2^{\aleph_0}$ and we study their properties. As an application, based on those families we construct a reflexive space $\mathfrak{X}_{2^{\aleph_0}}^α$, $α<ω_1$ with density the continuum, such that every bounded non norm convergent sequence $\{x_k\}_k$ has a subsequence generating $\ell_1^α$ as a spreading model.

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arXiv2013

A reflexive HI space with the hereditary Invariant Subspace Property

A reflexive hereditarily indecomposable Banach space $\mathfrak{X}_{_{^\text{ISP}}}$ is presented, such that for every $Y$ infinite dimensional closed subspace of $\mathfrak{X}_{_{^\text{ISP}}}$ and every bounded linear operator $T:Y\rightarrow Y$, the operator $T$ admits a non-trivial closed invariant subspace.

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arXiv2012

A type (4) space in (FR)-classification

We present a reflexive Banach space with an unconditional basis which is quasi-minimal and tight by range, i.e. of type (4) in Ferenczi-Rosendal list within the framework of Gowers' classification program of Banach spaces. The space is an unconditional variant of the Gowers Hereditarily Indecomposable space with asymptotically unconditional basis.

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arXiv2011

On spaces admitting no $\ell_p$ or $c_0$ spreading model

It is shown that for each separable Banach space $X$ not admitting $\ell_1$ as a spreading model there is a space $Y$ having $X$ as a quotient and not admitting any $\ell_p$ for $1 \leq p < \infty$ or $c_0$ as a spreading model. We also include the solution to a question of W.B. Johnson and H.P. Rosenthal on the existence of a separable space not admitting as a quotient any space with separable dual.

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