arXiv subjects
Publications and source records attributed to Spiros A. Argyros.
It is shown that for every $k\in\mathbb{N}$ and every spreading sequence $\{e_n\}_{n\in\mathbb{N}}$ that generates a uniformly convex Banach space $E$, there exists a uniformly convex Banach space $X_{k+1}$ admitting $\{e_n\}_{n\in\mathbb{N}}$ as a $k+1$-iterated spreading model, but not as a $k$-iterated one.
It is shown that variants of the HI methods could yield objects closely connected to the classical Banach spaces. Thus we present a new $c_0$ saturated space, denoted as $\mathfrak{X}_0$, with rather tight structure. The space $\mathfrak{X}_0$ is not embedded into a space with an unconditional basis and its complemented subspaces have the following structure. Everyone is either of type I, namely, contains an isomorph of $\mathfrak{X}_0$ itself or else is isomorphic to a subspace of $c_0$ (type II). Furthermore for any analytic decomposition of $\mathfrak{X}_0$ into two subspaces one is of type I and the other is of type II. The operators of $\mathfrak{X}_0$ share common features with those of HI spaces.
It is shown that every separable reflexive Banach space is a quotient of a reflexive Hereditarily Indecomposable space, which yields that every separable reflexive Banach is isomorphic to a subspace of a reflexive Indecomposable space. Furthermore, every separable reflexive Banach space is a quotient of a reflexive complementably $\ell_p$ saturated space with $1<p<\infty$ and of a $c_0$ saturated space.
We construct a weak Hilbert Banach space such that for every block subspace $Y$ every bounded linear operator on Y is of the form D+S where S is a strictly singular operator and D is a diagonal operator. We show that this yields a weak Hilbert space whose block subspaces are not isomorphic to any of their proper subspaces.
In the first part of the paper we present and discuss concepts of local and asymptotic hereditary proximity to \ell_1. The second part is devoted to a complete separation of the hereditary local proximity to \ell_1 from the asymptotic one. More precisely for every countable ordinal ξwe construct a separable reflexive space \mathfrak{X}_ξsuch that every infinite dimensional subspace of it has Bourgain \ell_1-index greater than ω^ξand the space itself has no \ell_1-spreading model. We also present a reflexive HI space admitting no \ell_p as a spreading model.
We provide a characterization of the Banach spaces $X$ with a Schauder basis $(e_n)_{n\in\mathbb{N}}$ which have the property that the dual space $X^*$ is naturally isomorphic to the space $\mathcal{L}_{diag}(X)$ of diagonal operators with respect to $(e_n)_{n\in\mathbb{N}}$ . We also construct a Hereditarily Indecomposable Banach space ${\mathfrak X}_D$ with a Schauder basis $(e_n)_{n\in\mathbb{N}}$ such that ${\mathfrak X}^*_D$ is isometric to $\mathcal{L}_{diag}({\mathfrak X}_D)$ with these Banach algebras being Hereditarily Indecomposable. Finally, we show that every $T\in \mathcal{L}_{diag}({\mathfrak X}_D)$ is of the form $T=λI+K$, where $K$ is a compact operator.
We construct a Hereditarily Indecomposable Banach space $\eqs_d$ with a Schauder basis \seq{e}{n} on which there exist strictly singular non-compact diagonal operators. Moreover, the space $\mc{L}_{\diag}(\eqs_d)$ of diagonal operators with respect to the basis \seq{e}{n} contains an isomorphic copy of $\ell_{\infty}(\N)$.
In the present work we provide a variety of examples of HI Banach spaces containing no reflexive subspace and we study the structure of their duals as well as the spaces of their linear bounded operators. Our approach is based on saturated extensions of ground sets and the method of attractors.
The present work consists of three parts. In the first one we determine the prototypes of separable Rosenthal compacta and we provide a classification theorem. The second part concerns an extension of a theorem of S. Todorcevic. The last one is devoted to applications.
It is shown that every Banach space either contains $\ell ^1$ or it has an infinite dimensional closed subspace which is a quotient of a H.I. Banach space.Further on, $L^p(λ)$, $1<p<\infty $, is a quotient of a H.I Banach space.
We study the modified and boundedly modified mixed Tsirelson spaces $T_M[({\cal F}_{k_n},θ_n)_{n=1}^{\infty }]$ and $T_{M(s)}[({\cal F}_{k_n},θ_n)_{n=1}^{\infty }]$ respectively, defined by a subsequence $({\cal F}_{k_n})$ of the sequence of Schreier families $({\cal F}_n)$. These are reflexive asymptotic $\ell_1$ spaces with an unconditio- nal basis $(e_i)_i$ having the property that every sequence $\{ x_i\}_{i=1}^n$ of normalized disjointly supported vectors contained in $\langle e_i\rangle_{i=n}^{\infty }$ is equivalent to the basis of $\ell_1^n$. We show that if $\limθ_n^{1/n}=1$ then the space $T[({\cal F}_n,θ_n) _{n=1}^{\infty }]$ and its modified variations are totally incomparable by proving that $c_0$ is finitely disjointly representable in every block subspace of $T[({\cal F}_n, θ_n)_{n=1}^{\infty }]$. Next, we present an example of a boundedly modified mixed Tsirelson space $X_{M(1),u}$ which is arbitrarily distortable. Finally, we construct a variation of the space $X_{M(1),u}$ which is hereditarily indecomposable.
It is proved that every normalized weakly null \sq\ has a sub\sq\ which is convexly unconditional. Further, an Hierarchy of summability methods is introduced and with this we give a complete classification of the complexity of weakly null \sq s.
Two examples of asymptotic $\ell_{1}$ Banach spaces are given. The first, $X_{u}$, has an unconditional basis and is arbitrarily distortable. The second, $X$, does not contain any unconditional basic sequence. Both are spaces of the type of Tsirelson. We thus answer a question raised by W.T.Gowers.
The class of countably intersected families of sets is defined. For any such family we define a Banach space not containing $\ell^{1}(\NN )$. Thus we obtain counterexamples to certain questions related to the heredity problem for W.C.G. Banach spaces. Among them we give a subspace of a W.C.G. Banach space not containing $\ell^{1}(\NN )$ and not being itself a W.C.G. space.
To any pair ( M , theta ) where M is a family of finite subsets of N compact in the pointwise topology, and 0 {1/n} then T_M^theta is reflexive. Moreover, if the Cantor-Bendixson index of M is greater than omega then T_M^theta does not contain any l^p, while if the Cantor-Bendixson index of M is finite thenT_M^theta contains some l^p or c_o . In particular, if M ={ A subset N : |A| leq n } and {1/n}<theta <1 then T_M^theta is isomorphic to some l^p .
A non RNP Banach space E is constructed such that $E^{*}$ is separable and RNP is equivalent to PCP on the subsets of E.