Polynomials on Schreier's space

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。

James-Schreier 空间与 Lorentz 空间的等距映射

Isometries of James-Schreier and Lorentz spaces

本文通过极值点保持方法研究实 Banach 序列空间上的满射线性等距群，刻画了 James-Schreier 空间 $V_1$ 对偶单位球的极值点，证明其等距映射仅为 $\pm Id$，并给出 Lorentz 序列空间的 Banach-Stone 型结果。

Baernstein 空间和 Schreier 空间上算子的闭理想

Closed ideals of operators on the Baernstein and Schreier spaces

本文研究Baernstein空间$B_p$和Schreier空间$S_p$上有界算子格的闭理想，证明在每个空间上均存在$2^{\mathfrak{c}}$个介于紧算子与严格奇异算子理想之间的闭理想，及$2^{\mathfrak{c}}$个包含无限秩投影的闭理想。

严格奇异算子在Baernstein、Schreier和ℓ_p空间直和上的紧性

Compactness of compositions of strictly singular operators on direct sums of Baernstein, Schreier and $\ell_p$-spaces

研究严格奇异算子在特定Banach空间直和上的紧性性质，证明商代数的nilpotent性及复合算子的紧性特征。

论无条件性与高阶Schreier无条件性

On Unconditionality and Higher-Order Schreier Unconditionality

该研究在泛函分析领域，提出比连续统假设更弱的条件A，证明满足该条件时，对所有可数序数α均为S_α-无条件的Banach空间基必为无条件基，深化了基的无条件性与高阶Schreier族相关理论。

On the geometry of higher order Schreier spaces

For each countable ordinal $α$ let $\mathcal{S}_α$ be the Schreier set of order $α$ and $X_{\mathcal{S}_α}$ be the corresponding Schreier space of order $α$. In this paper we prove several new properties of these spaces. 1) If $α$ is non-zero then $X_{\mathcal{S}_α}$ possesses the $λ$-property of R. Aron and R. Lohman and is a $(V)$-polyhedral spaces in the sense on V. Fonf and L. Vesely. 2) If $α$ is non-zero and $1<p<\infty$ then the $p$-convexification $X^{p}_{\mathcal{S}_α}$ possesses the uniform $λ$-property of R. Aron and R. Lohman. 3) For each countable ordinal $α$ the space $X^*_{\math…

Closed ideals of operators on the Tsirelson and Schreier spaces

Let $\mathscr{B}(X)$ denote the Banach algebra of bounded operators on $X$, where~$X$ is either Tsirelson's Banach space or the Schreier space of order $n$ for some $n\in\mathbb N$. We show that the lattice of closed ideals of~$\mathscr{B}(X)$ has a very rich structure; in particular $\mathscr{B}(X)$ contains at least continuum many maximal ideals. Our approach is to study the closed ideals generated by the basis projections. Indeed, the unit vector basis is an unconditional basis for each of the above spaces, so there is a basis projection $P_N\in\mathscr{B}(X)$ corresponding to each non-empt…

On quotients of Banach spaces having shrinking unconditional bases

It is proved that if a Banach space $Y$ is a quotient of a Banach space having a shrinking unconditional basis, then every normalized weakly null sequence in $Y$ has an unconditional subsequence. The proof yields the corollary that every quotient of Schreier's space is $c_o$-saturated.

Isometries of combinatorial Banach spaces

We prove that every isometry between two combinatorial spaces is determined by a permutation of the canonical unit basis combined with a change of signs. As a consequence, we show that in the case of Schreier spaces, all the isometries are given by a change of signs of the elements of the basis. Our results hold for both the real and the complex cases.

高阶Tsirelson空间与其修正版本是同构的

Higher Order Tsirelson Spaces and their Modified Versions are Isomorphic

该研究证明可数序数阶的Tsirelson空间$T_\xi$与其修正版通过恒等映射3-同构，同时证实对应阶Schreier族与其修正版相同，解决了相关公开问题，并应用该结果得出$T_\xi$上有界线性算子代数有$2^{\mathfrak c}$个闭理想。

The $ξ,ζ$-Dunford Pettis property

Using the hierarchy of weakly null sequences introduced by Argyros, Merkourakis, and Tsarpalias, we introduce two new families of operator classes. The first family simultaneously generalizes the completely continuous operators and the weak Banach-Saks operators. The second family generalizes the class $\mathfrak{DP}$. We study the distinctness of these classes, and prove that each class is an operator ideal. We also investigate the properties possessed by each class, such as injectivity, surjectivity, and identification of the dual class. We produce a number of examples, including the higher …

$α$-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 …

Banach spaces of polynomials as "large" subspaces of $\mathbf{\ell^\infty}$-spaces

In this note we study Banach spaces of traces of real polynomials on $\mathbb R^n$ to compact subsets equipped with supremum norms from the point of view of Geometric Functional Analysis.

Positive Polynomials on Riesz Spaces

We prove some properties of positive polynomial mappings between Riesz spaces, using finite difference calculus. We establish the polynomial analogue of the classical result that positive, additive mappings are linear. And we prove a polynomial version of the Kantorovich extension theorem.

组合巴拿赫空间的动物园

The ZOO of combinatorial Banach spaces

本文系统研究由任意有限集族以类 Schreier 方式诱导的组合巴拿赫空间，建立组合性质与几何性质之间的结构定理，并给出大量例子及不含 c₀ 拷贝子集族的复杂性分析。

Cyclic polynomials in Dirichlet-type spaces in the unit ball of $\mathbb C^2$

We characterize polynomials that are cyclic in Dirichlet-type spaces in the unit ball in $\mathbb C^2$

Polynomial approximation in weighted Dirichlet spaces

We give an elementary proof of an analogue of Fejér's theorem in weighted Dirichlet spaces with superharmonic weights. This provides a simple way of seeing that polynomials are dense in such spaces.