Subsymmetric sequences and minimal spaces

On Garling sequence spaces

The aim of this paper is to introduce and investigate a new class of separable Banach spaces modeled after an example of Garling from 1968. For each $1\leqslant p<\infty$ and each nonincreasing weight $\textbf{w}\in c_0\setminus\ell_1$ we exhibit an $\ell_p$-saturated, complementably homogeneous, and uniformly subprojective Banach space $g(\textbf{w},p)$. We also show that $g(\textbf{w},p)$ admits a unique subsymmetric basis despite the fact that for a wide class of weights it does not admit a symmetric basis. This provides the first known examples of Banach spaces where those two properties c…

Garling sequence spaces

By generalizing a construction of Garling, for each $1\leqslant p<\infty$ and each normalized, nonincreasing sequence of positive numbers $w\in c_0\setminus\ell_1$ we exhibit an $\ell_p$-saturated, complementably homogeneous Banach space $g(w,p)$ related to the Lorentz sequence space $d(w,p)$. Using methods originally developed for studying $d(w,p)$, we show that $g(w,p)$ admits a unique (up to equivalence) subsymmetric basis, although when $w=(n^{-θ})_{n=1}^\infty$ for some $0<θ<1$, it does not admit a symmetric basis. We then discuss some additional properties of $g(w,p)$ related to uniform …

Primarity of direct sums of Orlicz spaces and Marcinkiewicz spaces

Let $\mathbb{Y}$ be either an Orlicz sequence space or a Marcinkiewicz sequence space. We take advantage of the recent advances in the theory of factorization of the identity carried on in [R. Lechner, Subsymmetric weak* Schauder bases and factorization of the identity, arXiv:1804.01372 [math.FA]] to provide conditions on $\mathbb{Y}$ that ensure that, for any $1\le p\le\infty$, the infinite direct sum of $\mathbb{Y}$ in the sense of $\ell_p$ is a primary Banach space, enlarging this way the list of Banach spaces that are known to be primary.

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 i…

Baernstein 空间与 Schreier 空间上的严格奇异算子

Strictly singular operators on the Baernstein and Schreier spaces

该文证明在 Baernstein 与 Schreier 空间上，两个严格奇异算子复合为紧算子，并刻画了次对称基的等价性及非补块序列的存在。

Positional graphs and conditional structure of weakly null sequences

We prove that, unless assuming additional set theoretical axioms, there are no reflexive space without unconditional sequences of density the continuum. We give for every integer $n$ there are normalized weakly-null sequences of length $\om_n$ without unconditional subsequences. This together with a result of \cite{Do-Lo-To} shows that $\om_ω$ 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 $\ka$ which is smaller than the first $\om$-Erdös cardina…

关于Banach空间条件基对称类型的注记

Remarks about symmetry-type conditions of conditional bases of Banach spaces

该研究探讨了准Banach空间中条件基的对称性和扩散性，通过重范数构造使排列和递增映射成为等距映射，扩展了经典等距重范数定理的应用范围。

On the minimal Sums of sequences in the tensor product of separable Hilbert spaces

It is known that the tensor product of two sequences, in the tensor product of two separable Hilbert spaces, is a frame if and only if each component of that product is a frame. This paper proposes a sort of generalization of the aforementioned result by dealing with sequences S that are finite minimal sums of tensor products of a finite number of sequences. We prove that S is a Bessel sequence if and only if it is a sum for which each term is the tensor product of Bessel sequences. We also state necessary conditions for S to be a frame. For dimensions higher than one, we deduce several result…

On the minimal space problem and a new result on existence of basic sequences in quasi-Banach spaces

We prove that if $X$ is a quasi-normed space which possesses an infinite countable dimensional subspace with a separating dual, then it admits a strictly weaker Hausdorff vector topology. Such a topology is constructed explicitly. As an immediate consequence, we obtain an improvement of a well-known result of Kalton-Shapiro and Drewnowski by showing that a quasi-Banach space contains a basic sequence if and only if it contains an infinite countable dimensional subspace whose dual is separating. We also use this result to highlight a new feature of the minimal quasi-Banach space constructed by …

The Banach space S is complementably minimal and subsequentially prime

We first include a result of the second author showing that the Banach space S is complementably minimal. We then show that every block sequence of the unit vector basis of S has a subsequence which spans a space isomorphic to its square. By the Pełczyński decomposition method it follows that every basic sequence in S which spans a space complemented in S has a subsequence which spans a space isomorphic to S (i.e. S is a subsequentially prime space).

Subsequential minimality in Gowers and Maurey spaces

We define block sequences $(x_n)$ in every block subspace of a variant of the space of Gowers and Maurey so that the map $x_{2n-1}\mapsto x_{2n} $ extends to an isomorphism. This implies the existence of a subsequentially minimal HI space, which solves a question in \cite{FR}.

Weakly Radon-Nikodým Boolean algebras and independent sequences

A compact space is said to be weakly Radon-Nikodým (WRN) if it can be weak*-embedded into the dual of a Banach space not containing $\ell_1$. We investigate WRN Boolean algebras, i.e. algebras whose Stone space is WRN compact. We show that the class of WRN algebras and the class of minimally generated algebras are incomparable. In particular, we construct a minimally generated nonWRN Boolean algebra whose Stone space is a separable Rosenthal compactum, answering in this way a question of W. Marciszewski. We also study questions of J. Rodríguez and R. Haydon concerning measures and the existenc…

Minimal sequences and the Kadison-Singer problem

The Kadison-Singer problem asks: does every pure state on the diagonal sublgebra of the C*-algebra of bounded operators on a separable infinite dimensional Hilbert space admit a unique extension? A yes answer is equivalent to several open conjectures including Feichtinger's: every bounded frame is a finite union of Riesz sequences. We consider the special case: Feichtinger's conjecture for exponentials and prove that the set of projections onto a measurable subset of the circle group of the set of exponential functions equals a union of a finite number of Reisz sequences if and only if there e…

Minimality, homogeneity and topological 0-1 laws for subspaces of a Banach space

If a Banach space is saturated with basic sequences whose linear span embeds into the linear span of any subsequence, then it contains a minimal subspace. It follows that any Banach space is either ergodic or contains a minimal subspace. For a Banach space $X$ with an (unconditional) basis, topological 0-1 law type dichotomies are stated for block-subspaces of $X$ as well as for subspaces of $X$ with a successive FDD on its basis. A uniformity principle for properties of block-sequences, results about block-homogeneity, and a possible method to construct a Banach space with an unconditional ba…

A stability problem for some complete and minimal Gabor systems in $L^2(\mathbb{R})$

A Gabor system in $L^2(\mathbb{R})$, generated by a window $g\in L^2(\mathbb{R})$ and associated with a sequence of times and frequencies $Γ\subset\mathbb{R}^2$, is a set formed by translations in time and modulations of $g$. In this paper we consider the case when $g$ is the Gaussian function and $Γ$ is a sequence whose associated Gabor system $\mathcal{G}_Γ$ is complete and minimal in $L^2(\mathbb{R})$. We consider two main cases: that of the lattice without one point and that of the sequence constructed by Ascensi, Lyubarskii and Seip lying on the union of the coordinate axes of the time-fr…