Equivalence after extension and matricial coupling coincide with Schur coupling, on separable Hilbert spaces

Tsirelson-like spaces and complexity of classes of Banach spaces

Employing a construction of Tsirelson-like spaces due to Argyros and Deliyanni, we show that the class of all Banach spaces which are isomorphic to a subspace of $c_{0}$ is a complete analytic set with respect to the Effros Borel structure of separable Banach spaces. Moreover, the classes of all separable spaces with the Schur property and of all separable spaces with the Dunford-Pettis property are $\mathbfΠ^{1}_{2}$-complete.

马祖尔的可分商问题对于非可分博里森-皮塞尔L_∞空间

On $c_0$ quotients of nonseparable Bourgain-Pisier $\mathscr{L}_\infty$-spaces

本文解决马祖尔问题，证明满足特定条件的L_∞空间存在c_0商，通过构造性方法和反例验证假设的必要性。

Locally convex spaces and Schur type properties

In the main result of the paper we extend Rosenthal's characterization of Banach spaces with the Schur property by showing that for a quasi-complete locally convex space $E$ whose separable bounded sets are metrizable the following conditions are equivalent: (1) $E$ has the Schur property, (2) $E$ and $E_w$ have the same sequentially compact sets, where $E_w$ is the space $E$ with the weak topology, (3) $E$ and $E_w$ have the same compact sets, (4) $E$ and $E_w$ have the same countably compact sets, (5) $E$ and $E_w$ have the same pseudocompact sets, (6) $E$ and $E_w$ have the same functionall…

保守离散定常系统与块算子CMV矩阵

Conservative discrete time-invariant systems and block operator CMV matrices

众所周知，来自Schur类${\bf S}(\mathfrak M,\mathfrak N)$的算子值函数$Θ$（其中$\mathfrak M$和$\mathfrak N$是可分离的Hilbert空间）可以被实现为简单保守离散时不变线性系统的传递函数。已知的实现涉及函数$Θ$本身、Hardy空间或再生核Hilbert空间。另一方面，与经典纯量情形一样，Schur类算子值函数由其所谓的Schur参数唯一确定。在本文中，我们只使用舒尔参数构造简单的保守实现。结果表明，该系统所对应的酉算子是五对角块算子矩阵，它是单位圆上标量正交多项式理论中出现的Canadian-Moral-Velázquez（CMV）矩阵的类似物.对于任意完全非酉压缩，我们得到了由截断块算子CMV矩阵给出的新模型。证明了Hilbert空间中压缩算子的最小酉伸缩和单位圆上半谱算子测度的最小Naimark伸缩也可用块算子CMV矩阵表示.

Schur类算子值函数的迭代及其保守实现

Iterates of the Schur class operator-valued function and their conservative realizations

设$\mathfrak M$和$\mathfrak N$是可分的Hilbert空间，$Θ(λ)$是单位圆盘上全纯压缩函数Schur类${\bf S}(\mathfrak M,\mathfrak N)$中的一个函数.经典Schur算法的算子推广与$Θ$的压缩序列相关联（$Θ$的Schur参数）$Γ_0=Θ(0)\in \bL(\sM,\sN), Γ_n\in\bL(\sD_{Γ_{n-1}}, \sD_{Γ^*_{n-1}}) $和函数序列$Θ_0 = Θ$，$Θ_n\in {\bf S}(\sD_{Γ_n},\sD_{Γ^*_n})$ $ n=1,...$（$Θ$的Schur迭代）由关系\[ Γ_n=Θ_n(0), Θ_n(λ) = Γ_n+λD_{Γ^*_n} Θ_{n+1}(λ) (I + λΓ^*_nΘ_{n+1} (λ))^{-1}D_{Γ_n}, |λ|<1. \]连接。函数$Θ(λ)\in {\bf S}(\sM,\sN)$可以实现为线性保守和简单离散的传递函数\[ Θ(λ)=D+λC(I-λA)^{-1}B \]。时间系统$τ= {\begin{bmatrix}D & C \cr B & A\end{bmatrix}; \mathfrak M, \mathfrak N,\mathfrak H}$，其分别具有状态空间$\mathfrak H$以及输入和输出空间$\mat…

$ξ$-completely continuous operators and $ξ$-Schur Banach spaces

For each ordinal $0\leqslant ξ\leqslant ω_1$, we introduce the notion of a $ξ$-completely continuous operator and prove that for each ordinal $0< ξ< ω_1$, the class $\mathfrak{V}_ξ$ of $ξ$-completely continuous operators is a closed, injective operator ideal which is not surjective, symmetric, or idempotent. We prove that for distinct $0\leqslant ξ, ζ\leqslant ω_1$, the classes of $ξ$-completely continuous operators and $ζ$-completely continuous operators are distinct. We also introduce an ordinal rank $\textsf{v}$ for operators such that $\textsf{v}(A)=ω_1$ if and only if $A$ is completely co…

On quantification of weak sequential completeness

We consider several quantities related to weak sequential completeness of a Banach space and prove some of their properties in general and in $L$-embedded Banach spaces, improving in particular an inequality of G. Godefroy, N. Kalton and D. Li. We show some examples witnessing natural limits of our positive results, in particular, we construct a separable Banach space $X$ with the Schur property that cannot be renormed to have a certain quantitative form of weak sequential completeness, thus providing a partial answer to a question of G. Godefroy.

New spaces of matrices with operator entries

In this paper, we will consider matrices with entries in the space of operators $\mathcal{B}(H)$, where $H$ is a separable Hilbert space and consider the class of matrices that can be approached in the operator norm by matrices with a finite number of diagonals. We will use the Schur product with Toeplitz matrices generated by summability kernels to describe such a class and show that in the case of Toeplitz matrices it can be identified with the space of continuous functions with values in $\mathcal B(H)$. We shall also introduce matriceal versions with operator entries of classical spaces of…

On the complexity of some classes of Banach spaces and non-universality

These notes are dedicated to the study of the complexity of several classes of separable Banach spaces. We compute the complexity of the Banach-Saks property, the alternating Banach-Saks property, the complete continuous property, and the LUST property. We also show that the weak Banach-Saks property, the Schur property, the Dunford-Pettis property, the analytic Radon-Nikodym property, the set of Banach spaces whose set of unconditionally converging operators is complemented in its bounded oper- ators, the set of Banach spaces whose set of weakly compact operators is complemented in its bounde…

一致凸茎与库彻的$\ell_{\infty}(E)$问题

Uniformly convex stalks and Kucher's $\ell_{\infty}(E)$ problem

针对库彻提出的每个超自反巴拿赫空间E对应的ℓ_∞(E)是否为格罗滕迪克空间的问题，通过证明ℓ_∞(E)具有佩尔钦斯基性质(V)给出肯定回答，还推导了该空间上算子的若干推论。

A class of weakly compact sets in Lebesgue-Bochner spaces

Let $X$ be a Banach space and $μ$ a probability measure. A set $K \subseteq L^1(μ,X)$ is said to be a $δ\mathcal{S}$-set if it is uniformly integrable and for every $δ>0$ there is a weakly compact set $W \subseteq X$ such that $μ(f^{-1}(W)) \geq 1-δ$ for every $f\in K$. This is a sufficient, but in general non necessary, condition for relative weak compactness in $L^1(μ,X)$. We say that $X$ has property ($δ\mathcal{S}_μ$) if every relatively weakly compact subset of $L^1(μ,X)$ is a $δ\mathcal{S}$-set. In this paper we study $δ\mathcal{S}$-sets and Banach spaces having property ($δ\mathcal{S}_μ…