Emmanuel Fricain, Andreas Hartmann, William T. Ross, Dan Timotin
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.
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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。
This paper deals with the boundary behavior of functions in the de Branges--Rovnyak spaces. First, we give a criterion for the existence of radial limits for the derivatives of functions in the de Branges--Rovnyak spaces. This criterion generalizes a result of Ahern-Clark. Then we prove that the continuity of all functions in a de Branges--Rovnyak space on an open arc $I$ of the boundary is enough to ensure the analyticity of these functions on $I$. We use this property in a question related to Bernstein's inequality.
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。
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。
Let $\mathcal{H}(b)$ denote the de Branges--Rovnyak space associated with a function $b$ in the unit ball of $H^\infty(\mathbb{C}_+)$. We study the boundary behavior of the derivatives of functions in $\mathcal{H}(b)$ and obtain weighted norm estimates of the form $\|f^{(n)}\|_{L^2(μ)} \le C\|f\|_{\mathcal{H}(b)}$, where $f \in \mathcal{H}(b)$ and $μ$ is a Carleson-type measure on $\mathbb{C}_+\cup\mathbb{R}$. We provide several applications of these inequalities. We apply them to obtain embedding theorems for $\mathcal{H}(b)$ spaces. These results extend Cohn and Volberg--Treil embedding theo…
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…
We study reproducing kernel Hilbert and Pontryagin spaces of slice hyperholomorphic functions which are analogs of the Hilbert spaces of analytic functions introduced by de Branges and Rovnyak. In the first part of the paper we focus on the case of Hilbert spaces, and introduce in particular a version of the Hardy space. Then we define Blaschke factors and Blaschke products and we consider an interpolation problem. In the second part of the paper we turn to the case of Pontryagin spaces. We first prove some results from the theory of Pontryagin spaces in the quaternionic setting and, in partic…
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 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…
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。
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…
In this paper, we give an integral representation for the boundary values of derivatives of functions of the de Branges--Rovnyak spaces $\HH(b)$, where $b$ is in the unit ball of $H^\infty(\CC_+)$. In particular, we generalize a result of Ahern--Clark obtained for functions of the model spaces $K_b$, where $b$ is an inner function. Using hypergeometric series, we obtain a nontrivial formula of combinatorics for sums of binomial coefficients. Then we apply this formula to show the norm convergence of reproducing kernel $k_{ω,n}^b$ of the evaluation of $n$-th derivative of elements of $\HH(b)$ a…
Automorphic de Branges - Rovnyak Spaces
该研究针对模长≤1的β自守解析函数w，构建了依赖任意特征α的自守de Branges-Rovnyak空间族，通过格林函数乘运算定义酉算子，将其应用于Nevanlinna-Pick问题以扩展等距算子族。
In most classical holomorphic function spaces on the unit disk in which the polynomials are dense, a function $f$ can be approximated in norm by its dilates $f_r(z):=f(rz)~(r<1)$, in other words, $\lim_{r\to1^-}\|f_r-f\|=0$. We construct a de Branges-Rovnyak space ${\mathcal H}(b)$ in which the polynomials are dense, and a function $f\in{\mathcal H}(b)$ such that $\lim_{r\to1^-}\|f_r\|_{{\mathcal H}(b)}=\infty$. The essential feature of our construction lies in the fact that $b$ is an outer function.
A pathological de Branges-Rovnyak space
本文研究单位圆盘上的一类病态全纯函数空间，其中存在函数虽可用多项式逼近，但无法用其泰勒级数部分和逼近，并由此引出一个仍未解决的级数收敛问题。
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。
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。
Composition operators between de Branges-Rovnyak and Hardy spaces
研究从de Branges-Rovnyak空间到Hardy空间的复合算子的紧性，通过切片函数的Nevanlinna计数函数给出自然限制条件。
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。
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…
A counterexample concerning the coefficient formula for de Branges-Rovnyak norms
针对de Branges-Rovnyak空间的系数公式，本文构造反例，证明存在非极端点b对应的空间，其φ属于∩_{p<1}H^p，但H²中满足级数有限的函数不属于该空间，完善了此前相关结论。
Embeddings into de Branges-Rovnyak spaces
本文研究解析函数空间嵌入 de Branges-Rovnyak 空间的条件，通过乘子有界性准则刻画 Hardy 空间和 Dirichlet 空间的包含关系，并改进 Sarason 的端点结果。
Using the notion of commutative operator vessels, this work investigates de Branges-Rovnyak spaces whose elements are sections of a line bundle of multiplicative half-order differentials on a compact real Riemann surface. As a special case, we obtain a Beurling-Lax type theorem in the setting of the corresponding Hardy space on a finite bordered Riemann surface.
In this paper we discuss direct and reverse Carleson measures for the de Branges-Rovnyak spaces $\mathcal{H}(b)$, mainly when b is a non-extreme point of the unit ball of $H^{\infty}$.
We prove various Beurling-Lax type theorems, when the classical backward-shift operator is replaced by a general resolvent operator associated with a rational function. We also study connections to the Cuntz relations. An important tool is a new representation result for analytic functions, in terms of composition and multiplication operators associated with a given rational function. Applications to the theory of de Branges-Rovnyak spaces, also in the indefinite metric setting, are given.
In this paper we give an explicit description of de Branges-Rovnyak spaces $\HH(b)$ when $b$ is of the form $q^{r}$, where $q$ is a rational outer function in the closed unit ball of $H^{\infty}$ and $r$ is a positive number.
We study the invariant subspaces generated by inner functions for a class of $\mathcal{P}^t(μ)$-spaces which can be identified as spaces of analytic functions in the unit disk $\mathbb{D}$, where $μ$ is a measure supported in the closed unit disk and $\mathcal{P}^t(μ)$ is the span of analytic polynomials in the usual Lebesgue space $L^t(μ)$. Our measures define a range of spaces somewhere in between the Hardy and the Bergman spaces, and our results are thus a mixture of results from these two theories. For a large class of measures $μ$ we characterize the cyclic inner functions, and exhibit so…
A seminal result of Agler proves that the natural de Branges-Rovnyak kernel function associated to a bounded analytic function on the bidisk can be decomposed into two shift-invariant pieces. Agler's decomposition is non-constructive-a problem remedied by work of Ball-Sadosky-Vinnikov, which uses scattering systems to produce Agler decompositions through concrete Hilbert space geometry. This method, while constructive, so far has not revealed the rich structure shown to be present for special classes of functions-inner and rational inner functions. In this paper, we show that most of the impor…
Universal multipliers for Sub-Hardy Hilbert spaces
本文研究 de Branges-Rovnyak 空间的通用乘子，建立一般框架，重新证明 Davis-McCarthy 定理，并用商 b/a 所属的函数类刻画 Lipschitz 类与 Gevrey 类乘子。
Weighted Korenblum-Roberts Theory
本文解决加权Korenblum-Roberts问题，通过Thomson分解和关联Beurling-Carleson集合族，刻画了加权解析$\mathcal{P}^t(\mu)$空间中的循环奇异内函数，并应用于de Branges-Rovnyak空间的符号刻画。
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