Shuhei Kuwahara, Michio Seto
In this note, we give a new property of de Branges-Rovnyak kernels. As the main theorem, it is shown that the exponential of de Branges-Rovnyak kernel is strictly positive definite if the inner part of the corresponding Schur class function is nontrivial.
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We characterize the de Branges-Rovnyak spaces with complete Nevanlinna-Pick property. Our method relies on the general theory of reproducing kernel Hilbert spaces.
We prove that functions continuous up to the boundary are dense in de Branges-Rovnyak spaces induced by extreme points the unit ball of $H^\infty$.
The notes provide a short introduction to de Branges--Rovnyak spaces. They cover some basic facts and are intended to give the reader a taste of the theory, providing sufficient motivation to make it interesting.
We study a general metric constrained interpolation problem in a de Branges-Rovnyak space $\mathcal{H}(K_S)$ associated with a contractive multiplier $S$ between two Fock spaces along with its commutative counterpart, a de Branges-Rovnyak space associated with a Schur multiplier on the Drury-Arveson space of the unit ball of $\mathbb{C}^n$.
We study the compactness and the hypercyclicity of Toeplitz operators in the de Branges-Rovnyak spaces H(b) with co-analytic and bounded symbols on D. We highlight the fundamental role played by the function b generating the de Branges-Rovnyak space H(b). The characterization of compactness depends wether b is inner function or not and the characterization of hypercyclicity depends wether the function log(1-|b|) is integrable or not.
We show that de Branges-Rovnyak spaces include as special cases a number of spaces, such as the Hardy space, the Fock space, the Hardy-Sobolev space and the Dirichlet space. We present a general framework in which all these spaces can be obtained by specializing a sequence that appears in the construction. We show how to exploit this approach to solve interpolation problems in the Fock space.
A general interpolation problem with operator argument is studied for functions f from the de Branges-Rovnyak space H(s) associated with an analytic function s mapping the open unit disk D into the closed unit disk. The interpolation condition is taken in the Rosenblum-Rovnyak form f(A)c = b (with a suitable interpretation of f(A)c) for given Hilbert space operator A and two vectors b; c from the same space.
Nonextreme de Branges-Rovnyak spaces as models for contractions
已知de Branges-Rovnyak空间为希尔伯特空间上的压缩提供了一种替代的泛函模型，等价于Sz. Nagy-Foias模型。标量de Branges-Rovnyak空间$\mathcal{H}(b)$根据定义函数$b$是否在$H^\infty$的单位球内极值而具有本质上不同的性质.对于$b$极值，模型空间只是$\mathcal{H}(b)$，而对于$b$非极值，需要额外的构造。在本文中，我们确定的精确类的收缩作为一个模型$\mathcal{H}(b)$与$b$非极端。
In this paper, we characterize the boundedness, the compactness and the Hilbert-Schmidt property for composition operators acting from a de Branges-Rovnyak space $\mathcal H(b)$ into itself, when $b$ is a rational function in the closed unit ball of $H^\infty$ (but not a finite Blaschke product). In particular, we extend some of the results obtained by D. Sarason and J.N. Silva in the context of local Dirichlet spaces.
We describe de Branges-Rovnyak spaces $\mathcal H (b_α)$, $α>0$, where the function $b_α$ is not extreme in the unit ball of $H^{\infty}$ on the unit disk $\mathbb D$, defined by the equality $b_α(z)/a_α(z)=(1-z)^{-α}$, $z\in\mathbb D$, where $a_α$ is the outer function such that $a_α(0)>0$ and $|a_α|^2+|b_α|^2= 1$ a.e. on $\partial \mathbb D$.
Completely hyperexpansive operators with finite rank defect operator and de Branges-Rovnyak spaces
本文利用Aleman模型，用de Branges-Rovnyak空间的符号刻画了具有有限秩亏算子的循环、解析、完全超扩张算子。
Decompositions of de Branges-Rovnyak spaces
该研究针对内函数$I$对应的两类de Branges-Rovnyak空间，推导得出其正交分解形式，并探讨了这类空间的循环性相关问题。
Reproducing kernels, de Branges-Rovnyak spaces, and norms of weighted composition operators
证明了圆盘的Hardy空间H^2上的加权复合算子的范数是由与复合算子符号相关联的de Branges-Rovnyak空间中的权函数的范数控制的.作为推论，我们得到了H^2上复合算子有界性的一个新证明，并恢复了范数的标准上界。类似的论点适用于加权伯格曼空间。我们还证明了广义de Branges-Rovnyak核的正性足以证明复合算子在单位球上的标准函数空间上的有界性。
Löwner equations and de Branges--Rovnyak spaces
本文基于洛文纳理论，提出了一种新方法，用于在德布拉内-罗夫尼亚克空间中寻找具体元素。
Following ideas from the Abstract Interpolation Problem of Katsnelson et al. (Operators in spaces of functions and problems in function theory, vol 146, pp 83-69, Naukova Dumka, Keiv, 1987) for Schur class functions, we study a general metric constrained interpolation problem for functions from a vector-valued de Branges-Rovnyak space $\mathcal{H}(K_S)$ associated with an operator-valued Schur class function $S$. A description of all solutions is obtained in terms of functions from an associated de Branges-Rovnyak space satisfying only a bound on the de Branges-Rovnyak-space norm. Attention is…
In most classical holomorphic function spaces on the unit disk, a function $f$ can be approximated in the norm of the space by its dilates $f\_r(z):=f(rz)~(r \textless{} 1)$. We show that this is \emph{not} the case for the de Branges--Rovnyak spaces $\cH(b)$. More precisely, we give an example of a non-extreme point $b$ of the unit ball of $H^\infty$ and a function $f\in\cH(b)$ such that $\lim\_{r\to1^-}\|f\_r\|\_{\cH(b)}=\infty$. It is known that, if $b$ is a non-extreme point of the unit ball of $H^\infty$, then polynomials are dense in $\cH(b)$. We give the first constructive proof of this…
For $b\in H^\infty_1$, the closed unit ball of $H^\infty$, the de Branges-Rovnyak spaces $\mathcal{H}(b)$ is a Hilbert space contractively contained in the Hardy space $H^2$ that is invariant by the backward shift operator $S^*$. We consider the reducing subspaces of the operator $S^{*2}|_{\mathcal{H}(b)}$. When $b$ is an inner function, $S^{*2}|_{\mathcal{H}(b)}$ is a truncated Toepltiz operator and its reducibility was characterized by Douglas and Foias using model theory. We use another approach to extend their result to the case where $b$ is extreme. We prove that if $b$ is extreme but not…
Using an explicit construction, we show that the de Branges-Rovnyak spaces $\mathscr{H}(b)$, corresponding to rational $b$, are contained in their associated Smirnov classes.
Unbounded Toeplitz operators and finite rank de Branges-Rovnyak spaces
本文研究了有限秩德布兰斯-罗文纳克空间H(B)的函数理论方面，通过推广萨拉斯逊的工作，将有限秩H(B)-空间表征为托利齐亚算子T_φ*的伴随算子的域，并推导了H(B)-空间中函数的范数公式，同时应用该结果扩展了萨拉斯逊对经典德布兰斯-罗文纳克空间的判据。
Boundary behavior of functions in the de Branges--Rovnyak spaces
本文研究de Branges-Rovnyak空间中函数的边界表现。首先，我们给出了de Branges-Rovnyak空间中函数导数径向极限存在性的一个判据。该准则推广了Ahern-Clark的一个结果。然后证明了de Branges-Rovnyak空间中所有函数在边界的开弧$I$上的连续性足以保证这些函数在$I$上的解析性。我们在一个与伯恩斯坦不等式有关的问题中使用这个性质。
Higher order isometric shift operator on the de Branges-Rovnyak space
本文证明 de Branges-Rovnyak 空间可作为秩一亏差的扩张拟解析 2n-等距算子的模型空间，并刻画了其上 2n-等距前向移位算子的不变子空间。
The Cauchy dual subnormality problem (for short, CDSP) asks whether the Cauchy dual of a $2$-isometry is subnormal. In this paper, we address this problem for cyclic $2$-isometries. In view of some recent developments in operator theory on function spaces (see \cite{AM, LGR}), one may recast CDSP as the problem of subnormality of the Cauchy dual $\mathscr M'_z$ of the multiplication operator $\mathscr M_z$ acting on a de Branges-Rovnyak space $\mathcal H(B),$ where $B$ is a vector-valued rational function. The main result of this paper characterizes the subnormality of $\mathscr M'_z$ on $\mat…
De Branges-Rovnyak realizations of operator-valued Schur functions on the complex right half-plane
给出了复右半平面半平面上任意给定算子Schur函数的可控能量保持和可观的共能量保持的de Branges-Rovnyak泛函模型实现.为了揭示连续时间系统理论的技术细节，我们在半平面上完全地推导出这个理论，而不使用圆盘情形的结果。在文章的最后，我们明确的连接到相应的经典de Branges-Rovnyak实现的复单位圆盘上的舒尔函数。
de Branges-Rovnyak spaces: basics and theory
对于$S$，单位圆盘${\mathbb D}$上的压缩解析算子值函数，de Branges和Rovnyak将解析函数${\mathcal H}(S)$的希尔伯特空间和由单位圆盘${\mathbb D}$上的解析函数对组成的相关扩展空间${\mathcal D(S)}$关联起来。本综述描述了de Branges-Rovnyak空间${\mathcal H}(S)$的三个等价公式（原始几何de Branges-Rovnyak定义，Toeplitz算子特征，以及作为再生核Hilbert空间的特征），以及它作为完全非等距Hilbert空间收缩算子建模的基础希尔伯特空间的作用。还研究了这些想法的扩展，以处理建模的更一般的类完全非酉收缩算子，其中更一般的两个组件德布兰吉斯-Rovnyak模型空间${\mathcal D}(S)$和相关的重叠空间发挥关键作用。与其他函数论问题和应用的连接也进行了讨论。最近的应用到各种后续的应用程序中给出了一个配套的调查文章。
de Branges-Rovnyak spaces which are complete Nevanlinna-Pick spaces
本文刻画了 de Branges-Rovnyak 空间为完全 Nevanlinna-Pick 空间的条件，推广了 Hardy 空间和 Drury-Arveson 空间的已有结果，并证明双圆盘或单位圆盘情形仅当等距同构于 Hardy 空间时成立，而 $n\ge 3$ 时永不成立。
Cyclicity of the shift operator and a related completeness problem in de Branges-Rovnyak spaces
该文研究de Branges-Rovnyak空间上移位算子的循环向量，通过柯西变换建立其与完备性问题的关联，进而得到循环性相关的若干推论。
Orthogonal polynomials in de Branges--Rovnyak spaces
本文研究了正交多项式在de Branges-Rovnyak空间中的表示结构，并探讨了当b为有理函数时的线性系统相关结构及复合算子的不变性。
We show that there exists a de Branges-Rovnyak space $\mathcal{H}(b)$ on the unit disk containing a function $f$ with the following property: even though $f$ can be approximated by polynomials in $\mathcal{H}(b)$, neither the Taylor partial sums of $f$ nor their Cesàro, Abel, Borel or logarithmic means converge to $f$ in $\mathcal{H}(b)$. A key tool is a new abstract result showing that, if one regular summability method includes another for scalar sequences, then it automatically does so for certain Banach-space-valued sequences too.
Embedding model and de Branges-Rovnyak spaces in Dirichlet spaces
该研究利用再生核估计，刻画了de Branges-Rovnyak空间与调和加权Dirichlet空间、模型空间与局部Dirichlet空间的嵌入关系，并给出相关应用。
Schur multipliers and de Branges-Rovnyak spaces: the multiscale case
我们考虑有界线性算子作用在由齐次树的节点索引的$\ell_2$空间上。利用树上本原移位之间的Cuntz关系，推广了单尺度时变点赋值的概念，并引入了相应的再生核希尔伯特空间，柯西公式在其中成立.这些概念，然后用于研究舒尔乘数和相关的德布朗日-Rovnyak空间。作为应用，我们得到了Schur乘子作为多尺度输入-状态-输出系统传递算子的实现。
On the Divergence of Taylor Series in de Branges-Rovnyak Spaces
本文研究de Branges-Rovnyak空间中Taylor级数的发散性，证明部分和可在范数上趋于无穷，并将结果推广至Cesàro平均等求和法。
We consider Dirichlet spaces with superharmonic weights. This class contains both the harmonic weights and the power weights. Our main result is a characterization of the Dirichlet spaces with superharmonic weights that can be identified as de Branges-Rovnyak spaces. As an application, we obtain the dilation inequality \[ {\cal D}_ω(f_r)\le \frac{2r}{1+r}{\cal D}_ω(f) \qquad(0\le r<1), \] where ${\cal D}_ω$ denotes the Dirichlet integral with superharmonic weight $ω$, and $f_r(z):=f(rz)$ is the $r$-dilation of the holomorphic function $f$.
The Cowen-Douglas class and de Branges-Rovnyak spaces
本文建立了de Branges-Rovnyak空间与Cowen-Douglas算子类的联系，证明了后向移位算子在符号为极值点时绝不属于该类，在非极值时属于秩1的该类，并计算了曲率及得出酉等价和角导数的奇异结果。
Weighted composition operators on de Branges-Rovnyak spaces
本文研究了de Branges-Rovnyak空间上加权复合算子的有界性和紧性，通过与Hardy空间算子的关系扩展了相关研究成果。
We investigate expansive Hilbert space operators $T$ that are finite rank perturbations of isometric operators. If the spectrum of $T$ is contained in the closed unit disc $\overline{\mathbb{D}}$, then such operators are of the form $T= U\oplus R$, where $U$ is isometric and $R$ is unitarily equivalent to the operator of multiplication by the variable $z$ on a de Branges-Rovnyak space $\mathcal{H}(B)$. In fact, the space $\mathcal{H}(B)$ is defined in terms of a rational operator-valued Schur function $B$. In the case when $\dim \ker T^*=1$, then $\mathcal{H}(B)$ can be taken to be a space of …
Weighted norm inequalities for de Branges--Rovnyak spaces and their applications
设$\mathcal{H}(b)$表示与$H^\infty(\mathbb{C}_+)$的单位球中的函数$b$相关联的de Branges-Rovnyak空间。研究了$\mathcal{H}(b)$中函数导数的边界表现，得到了$\|f^{(n)}\|_{L^2(μ)} \le C\|f\|_{\mathcal{H}(b)}$形式的加权范数估计，其中$f \in \mathcal{H}(b)$和$μ$是$\mathbb{C}_+\cup\mathbb{R}$上的Carleson型测度.我们提供了这些不等式的几个应用。应用它们得到了$\mathcal{H}(b)$空间的嵌入定理。这些结果推广了Cohn和Volberg-Treil嵌入定理的模型（星不变）的子空间，这是特殊类的de Branges-Rovnyak空间。利用导数不等式研究了$\mathcal{H}(b)$中再生核$\{k^b_{λ_n}\}$的Riesz基在$λ_n$点的小扰动下的稳定性.
De Branges-Rovnyak spaces and local Dirichlet spaces of higher order
该研究围绕非极端有理函数生成的de Branges-Rovnyak空间与高阶局部Dirichlet空间展开，给出了高阶局部Dirichlet空间的导数刻画、对应生成函数的显式公式，以及相关移位算子限制的游荡向量性质。
In this paper, we study the cyclicity problem with respect to the forward shift operator $S_b$ acting on the de Branges--Rovnyak space $\mathscr{H}(b)$ associated to a function $b$ in the closed unit ball of $H^\infty$ and satisfying $\log(1-|b|)\in L^1(\mathbb T)$. We present a characterisation of cyclic vectors for $S_b$ when $b$ is a rational function which is not a finite Blaschke product. This characterisation can be derived from the description, given in [S. Luo, C. Gu, S. Richter, Higher order local Dirichlet integrals and de Branges--Rovnyak spaces, \emph{Adv. Math., \textbf{385} (2021…
Pontryagin de Branges Rovnyak spaces of slice hyperholomorphic functions
本文研究切片超全纯函数的再生核Hilbert空间和Pontryagin空间，它们是de Branges和Rovnyak引入的解析函数的希尔伯特空间的类似物.在本文的第一部分，我们专注于希尔伯特空间的情况下，并特别介绍了一个版本的哈代空间。然后定义了Blaschke因子和Blaschke乘积，并考虑了一个插值问题.在本文的第二部分，我们转向庞特里亚金空间的情况。我们首先证明了一些结果从理论的Pontryagin空间在四元数设置，特别是一个定理的Shmulyan稠密定义的压缩线性关系。然后研究了广义Schur函数和广义Carath'eodory函数的实现。
Littlewood subordination for de Branges--Rovnyak spaces
本文研究不同de Branges-Rovnyak空间之间的复合算子，受Mashreghi和Shabankhah启发，并解答了Dellepiane和Seco的开放问题，使用再生核方法、Sarason积分算子方法和Aleksandrov-Clark测度。
De Branges-Rovnyak spaces generated by row Schur functions with mate
本文研究由带伴偶的行 Schur 函数生成的 de Branges-Rovnyak 空间，证明多项式稠密性并刻画后移不变子空间，进而在有限秩条件下描述循环向量。
On Dirichlet-type and $n$-isometric shifts in finite rank de~Branges--Rovnyak spaces
本文研究有限秩 de Branges–Rovnyak 空间中 Dirichlet 型函数空间上的移位算子，证明有理符号与有限原子测度及有限秩亏格算子的等价性，并刻画了多参数情形下扩张移位算子的允许元组条件。
We present a solution of the operator-valued Schur-function realization problem on the right-half plane by developing the corresponding de Branges-Rovnyak canonical conservative simple functional model. This model corresponds to the closely connected unitary model in the disk setting, but we work the theory out directly in the right-half plane, which allows us to exhibit structure which is absent in the disk case. A main feature of the study is that the connecting operator is unbounded, and so we need to make use of the theory of well-posed continuous-time systems. In order to strengthen the c…
On the equality of De Branges-Rovnyak and Dirichlet spaces
本文完全刻画了当测度为有限原子和时，哪些 de Branges-Rovnyak 空间与调和加权 Dirichlet 空间相等，推广了 Costara-Ransford 的结果，无需对 Pythagorean 对作假设。
In this paper we begin the study of Schur analysis and de Branges-Rovnyak spaces in the framework of Fueter hyperholomorphic functions. The difference with other approaches is that we consider the class of functions spanned by Appell-like polynomials. This approach is very efficient from various points of view, for example in operator theory, and allows to make connections with the recently developed theory of slice polyanalytic functions. We tackle a number of problems: we describe a Hardy space, Schur multipliers and related results. We also discuss Blaschke functions, Herglotz multipliers a…
Let $B_n$ denote the unit ball of $\mathbb{C}^n$, $n\ge 1$, and let $\mathcal{D}$ denote a finite product of $B_{n_j}$, $j\ge 1$. Given a non-constant holomorphic function $b: \mathcal{D} \to B_1$, we study the corresponding family $σ_α[b]$, $α\in\partial B_1$, of Clark measures on the distinguished boundary $\partial\mathcal{D}$. We construct a natural unitary operator from the de Branges-Rovnyak space $\mathcal{H}(b)$ onto the Hardy space $H^2(σ_α)$. As an application, for $\mathcal{D}= B_n$ and an inner function $I: B_n \to B_1$, we show that the property $σ_1[I]\llσ_1[b]$ is directly relat…
Integral representation of the $n$-th derivative in de Branges-Rovnyak spaces and the norm convergence of its reproducing kernel
本文给出了de Branges-Rovnyak空间$\HH(b)$中函数导数边值的一个整表示，其中$b$在$H^\infty(\CC_+)$的单位球上.特别地，我们推广了Ahern-Clark关于模型空间$K_b$的函数的一个结果，其中$b$是内函数.利用超几何级数，得到了一个关于二项式系数和的非平凡组合数学公式。然后我们应用这个公式来显示再生核$k_{ω,n}^b$的规范收敛的评估$n$ -次导数的元素$\HH(b)$在点$ω$，因为它趋向于径向实轴的一个点。
On cyclicity in de Branges-Rovnyak spaces
本文研究 de Branges-Rovnyak 空间循环向量的刻画，将其归结为子空间 $(aH^2)^\perp$，在有限维情形给出完整函数论描述，并在无限维情形联系 $H^1$ 暴露点给出充分条件及基于 Aleksandrov-Clark 测度的必要条件。
Representation of simple symmetric operators with deficiency indices (1,1) in de Branges space
最近证明了任何具有亏指数（1，1）的正则单对称算子酉等价于具有Kramer采样性质的实直线上函数的再生核希尔伯特空间中的乘法算子.这项工作的动机，部分，潜在的应用信号处理和数学物理。本文利用de Branges-Rovnyak空间和对称算子的特征函数的已知结果，证明了任意对称算子在整函数的de Branges空间中与自变量相乘是酉等价的.这导致了关于这种对称算子的谱的简单的新结果，关于在上半平面的de Branges-Rovnyak空间中何时密集地定义乘以z的乘法，以及关于存在从$L^2（R，dv）的给定子空间到整函数的de Branges空间的等距的充分条件，该等距充当与可测函数的乘法。
A non-commutative de Branges-Rovnyak model for row contractions
本文提出非交换德布兰-罗文纳克模型，用于描述CNC行收缩的特征函数及其单位不变性。
For the class of de Branges-Rovnyak spaces $\mathcal{H}(b)$ of the unit disk $\mathbb{D}$ defined by extreme points $b$ of the unit ball of $H^\infty$, we study the problem of approximation of a general function in $\mathcal{H}(b)$ by a function with an extension to the unit circle $\mathbb{T}$ of some degree of smoothness, for instance satisfying Hölder estimates or being differentiable. We will exhibit connections between this question and the theory of subnormal operators and, in particular, we will tie the possibility of smooth approximations to properties of invariant subspaces of a certa…
Interpolation and random interpolation in de Branges-Rovnyak spaces
本文研究了de Branges-Rovnyak空间中通用和乘子插值序列的特征，并探讨了具有指定半径的随机插值序列的0-1定律，同时证明了分解在M(a)中的正交性。
On de Branges--Rovnyak Kernels Admitting a Complete Pick Factor
本文刻画了de Branges-Rovnyak核允许完全Pick因子的条件，推广了Ahmed、Das和Panja的框架，并应用于Chu和Luo-Zhu的经典结果。
Generalized de Branges-Rovnyak spaces
该研究推广了经典de Branges-Rovnyak空间，引入再生核为$k(1-bb^*)$的$\u210b_k(b)$空间，建立类Sz.-Nagy-Foiaş模型，解决了相关多项式逼近猜想。
We apply the theory of de Branges-Rovnyak spaces to describe kernels of some Toeplitz operators on the classical Hardy space $H^2$. In particular, we discuss the kernels of the operators $T_{\bar f/ f}$ and $T_{\bar I\bar f/ f}$, where $f$ is an outer function in $H^2$ and $I$ is inner such that $I(0)=0$. We also obtain results on de Branges-Rovnyak spaces generated by nonextreme functions.
Infinitely supported harmonically weighted Dirichlet spaces which are de Branges Rovnyak spaces
本文研究调和加权Dirichlet空间与de Branges-Rovnyak空间的相等问题，突破有限支撑限制，刻画了无限支撑测度下两类空间相等的条件，还建立了相关T(1)型结果与扰动定理，并证明了Brown-Shields猜想。
On the coefficient formula for de Branges-Rovnyak norms
本文扩展了 de Branges-Rovnyak 空间范数的系数公式适用范围，证明了当 φ 属于 H^2 或为有理函数时公式对所有使右边有限的 H^2 函数成立，并给出了对一般非极值 b 和所有 H(b) 中函数成立的极限形式。
We study triplets of Hilbert space operators satisfying a certain inequality. A range inclusion theorem with norm estimate for those triplets is given with the language of Kre\uın space geometry and de Branges-Rovnyak space theory.
In this paper we extend vessel theory, or equivalently, the theory of overdetermined $2D$ systems to the Pontryagin space setting. We focus on realization theorems of the various characteristic functions associated to such vessels. In particular, we develop an indefinite version of de Branges-Rovnyak theory over real compact Riemann surfaces. To do so, we use the theory of contractions in Pontryagin spaces and the theory of analytic kernels with a finite number of negative squares. Finally, we utilize the indefinite de Branges-Rovnyak theory on compact Riemann surfaces in order to prove a Beur…
In this note, we describe the backward shift invariant subspaces for a large class of reproducing kernel Hilbert spaces. This class includes in particular de Branges-Rovnyak spaces (the non-extreme case) and the range space of co-analytic Toeplitz operators.
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