De Branges Rovnyak Spaces Research Articles

Weighted Dirichlet spaces that are de Branges-Rovnyak spaces with equivalent norms

For a positive Borel measure μ in the unit disc, we examine which weighted Dirichlet spaces Dμ can be identified with de Branges-Rovnyak spaces Hb, with equivalent norms. We prove a necessary condition for the equality Dμ=Hb and we explore its consequences. For Carleson measures μ, we give a necessary and sufficient condition for the equality Dμ=Hbμ, for a certain outer function bμ related with the balayage of μ on the unit circle, and we provide examples of those spaces.

On the divergence of Taylor series in de Branges–Rovnyak spaces

It is known that there exist functions in certain de Branges–Rovnyak spaces whose Taylor series diverge in norm, even though polynomials are dense in the space. This is often proved by showing that the sequence of Taylor partial sums is unbounded in norm. In this note we show that it can even happen that the Taylor partial sums tend to infinity in norm. We also establish similar results for lower-triangular summability methods such as the Cesàro means.

De Branges-Rovnyak spaces generated by row Schur functions with mate

In this paper, we study the de Branges-Rovnyak spaces H(B) generated by row Schur functions B with mate a. We prove that the polynomials are dense in H(B), and characterize the backward shift invariant subspaces of H(B). We then describe the cyclic vectors in H(B) when B is of finite rank and dim⁡(aH2)⊥<∞.

Mixed de Branges–Rovnyak and sub-Bergman spaces

In this paper we obtain some general results about mixed second order de Branges–Rovnyak spaces defined by an operator A such that both A and A⁎ are 2-hypercontractions. As applications of these results, we study mixed second order sub-Bergman spaces since the analytic Toeplitz operator Tb on weighted Bergman space Aα2 for α≥0 on the unit disk is such that both Tb and Tb⁎ are 2-hypercontractions.

Embedding model and de Branges-Rovnyak spaces in Dirichlet spaces

In this paper we study embeddings between de Branges-Rovnyak spaces and harmonically weighted Dirichlet spaces in terms of the boundary spectrum of b and the support of the measure μ, by using elementary reproducing kernel estimates. We completely characterize the embedding between the model spaces and the local Dirichlet spaces , and we discuss some applications.

De Branges–Rovnyak spaces and local Dirichlet spaces of higher order

Abstract We discuss de Branges–Rovnyak spaces ℋ ⁢ ( b ) {\mathcal{H}(b)} generated by nonextreme and rational functions b and local Dirichlet spaces of order m introduced in [S. Luo, C. Gu and S. Richter, Higher order local Dirichlet integrals and de Branges–Rovnyak spaces, Adv. Math. 385 2021, Paper No. 107748]. In that paper, the authors characterized nonextreme b for which the operator Y = S | ℋ ⁢ ( b ) {Y=S|_{\mathcal{H}(b)}} , the restriction of the shift operator S on H 2 {H^{2}} to ℋ ⁢ ( b ) {\mathcal{H}(b)} , is a strict 2 ⁢ m {2m} -isometry and proved that such spaces ℋ ⁢ ( b ) {\mathcal{H}(b)} are equal to local Dirichlet spaces of order m. Here we give a characterization of local Dirichlet spaces of order m in terms of the m-th derivatives that is a generalization of a known result on local Dirichlet spaces. We also find explicit formulas for b in the case when ℋ ⁢ ( b ) {\mathcal{H}(b)} coincides with local Dirichlet space of order m with equality of norms. Finally, we prove a property of wandering vectors of Y analogous to the property of wandering vectors of the restriction of S to harmonically weighted Dirichlet spaces obtained in [D. Sarason, Harmonically weighted Dirichlet spaces associated with finitely atomic measures, Integral Equations Operator Theory 31 1998, 2, 186–213].

Composition operators on de Branges–Rovnyak spaces associated to a rational (not inner) function

In this paper, we characterize the boundedness, the compactness and the Hilbert–Schmidt property for composition operators acting from a de Branges–Rovnyak space H(b) into itself, when b is a rational function in the closed unit ball of H∞ (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.

Thomson decompositions of measures in the disk

We study the classical problem of identifying the structure of P 2 ( μ ) \mathcal {P}^2(\mu ) , the closure of analytic polynomials in the Lebesgue space L 2 ( μ ) L^2(\mu ) of a compactly supported Borel measure μ \mu living in the complex plane. In his influential work, Thomson [Ann. of Math. (2) 133 (1991), pp. 477–507] showed that the space decomposes into a full L 2 L^2 -space and other pieces which are essentially spaces of analytic functions on domains in the plane. For a family of measures μ \mu supported on the closed unit disk D ¯ \overline {\mathbb {D}} which have a part on the open disk D \mathbb {D} which is similar to the Lebesgue area measure, and a part on the unit circle T \mathbb {T} which is the restriction of the Lebesgue linear measure to a general measurable subset E E of T \mathbb {T} , we extend the ideas of Khrushchev and calculate the exact form of the Thomson decomposition of the space P 2 ( μ ) \mathcal {P}^2(\mu ) . It turns out that the space splits according to a certain natural decomposition of measurable subsets of T \mathbb {T} which we introduce. We highlight applications to the theory of the Cauchy integral operator and de Branges-Rovnyak spaces.

A Local Douglas formula for Higher Order Weighted Dirichlet-Type Integrals

We prove a local Douglas formula for higher order weighted Dirichlet-type integrals. With the help of this formula, we study the multiplier algebra of the associated higher order weighted Dirichlet-type spaces $${\mathcal {H}}_{\pmb \mu },$$ induced by an m-tuple $$\pmb \mu =(\mu _1,\ldots ,\mu _{m})$$ of finite non-negative Borel measures on the unit circle. In particular, it is shown that any weighted Dirichlet-type space of order m, for $$m\geqslant 3,$$ forms an algebra under pointwise product. We also prove that every non-zero closed $$M_z$$ -invariant subspace of $${\mathcal {H}}_{\pmb \mu },$$ has codimension 1 property if $$m\geqslant 3$$ or $$\mu _2$$ is finitely supported. As another application of this local Douglas formula obtained in this article, it is shown that for any $$m\geqslant 2,$$ weighted Dirichlet-type space of order m does not coincide with any de Branges–Rovnyak space $${\mathcal {H}}(b)$$ with equivalence of norms.

Cyclicity in de Branges–Rovnyak spaces

Abstract In this paper, we study the cyclicity problem with respect to the forward shift operator Sb acting on the de Branges–Rovnyak space ℋ (b) associated to a function b in the closed unit ball of H ∞ and satisfying log(1− |b| ∈ L 1(𝕋). We present a characterisation of cyclic vectors for Sb when b is a rational function which is not a finite Blaschke product. This characterisation can be derived from the description, given in [22], of invariant subspaces of Sb in this case, but we provide here an elementary proof. We also study the situation where b has the form b = (1+ I)/2, where I is a non-constant inner function such that the associated model space K I = ℋ (I) has an orthonormal basis of reproducing kernels.

One Component Bounded Functions

Three different characterizations of one-component bounded analytic functions are provided. The first one is related to the the inner-outer factorization, the second one is in terms of the size of the reproducing kernels in the corresponding de Branges–Rovnyak spaces and the last one concerns the associated Aleksandrov–Clark measure.

Constructions of some families of smooth Cauchy transforms

AbstractFor a given Beurling–Carleson subset E of the unit circle $\mathbb {T}$ which has positive Lebesgue measure, we give explicit formulas for measurable functions supported on E such that their Cauchy transforms have smooth extensions from $\mathbb {D}$ to $\mathbb {T}$ . The existence of such functions has been previously established by Khrushchev in 1978, in non-constructive ways by the use of duality arguments. We construct several families of such smooth Cauchy transforms and apply them in a few related problems in analysis: an irreducibility problem for the shift operator, an inner factor permanence problem. Our development leads to a self-contained duality proof of the density of smooth functions in a very large class of de Branges–Rovnyak spaces. This extends the previously known approximation results.

On the problem of smooth approximations in [formula omitted] and connections to subnormal operators

For the class of de Branges-Rovnyak spaces H(b) of the unit disk D defined by extreme points b of the unit ball of H∞, we study the problem of approximation of a general function in H(b) by a function with an extension to the unit circle 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 certain subnormal operator. This leads us to several computable conditions on b which are necessary for such approximations to be possible. For a large class of extreme points b we use our result to obtain explicit necessary and sufficient conditions on the symbol b which guarantee the density of functions with differentiable boundary values in the space H(b). These conditions include an interplay between the modulus of b on T and the spectrum of its inner factor.

Cyclic inner functions in growth classes and applications to approximation problems

AbstractIt is well known that for any inner function $\theta $ defined in the unit disk $\mathbb {D}$ , the following two conditions: (i) there exists a sequence of polynomials $\{p_n\}_n$ such that $\lim _{n \to \infty } \theta (z) p_n(z) = 1$ for all $z \in \mathbb {D}$ and (ii) $\sup _n \| \theta p_n \|_\infty &lt; \infty $ , are incompatible, i.e., cannot be satisfied simultaneously. However, it is also known that if we relax the second condition to allow for arbitrarily slow growth of the sequence $\{ \theta (z) p_n(z)\}_n$ as $|z| \to 1$ , then condition (i) can be met for some singular inner function. We discuss certain consequences of this fact which are related to the rate of decay of Taylor coefficients and moduli of continuity of functions in model spaces $K_\theta $ . In particular, we establish a variant of a result of Khavinson and Dyakonov on nonexistence of functions with certain smoothness properties in $K_\theta $ , and we show that the classical Aleksandrov theorem on density of continuous functions in $K_\theta $ is essentially optimal. We consider also the same questions in the context of de Branges–Rovnyak spaces $\mathcal {H}(b)$ and show that the corresponding approximation result also is optimal.

Power-Series Summability Methods in de Branges–Rovnyak Spaces

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.

Unbounded Multipliers of Complete Pick Spaces

We examine densely defined (but possibly unbounded) multiplication operators in Hilbert function spaces possessing a complete Nevanlinna–Pick (CNP) kernel. For such a densely defined operator T, the domains of T and \(T^*\) are reproducing kernel Hilbert spaces contractively contained in the ambient space. We study several aspects of these spaces, especially the domain of \(T^*\), which can be viewed as analogs of the classical deBranges–Rovnyak spaces in the unit disk.

Clark measures and de Branges–Rovnyak spaces in several variables

Let denote the unit ball of , , and let denote a finite product of , . Given a non-constant holomorphic function , we study the corresponding family , , of Clark measures on the distinguished boundary . We construct a natural unitary operator from the de Branges–Rovnyak space onto the Hardy space . As an application, for and an inner function , we show that the property is directly related to the membership of an appropriate explicit function in .

Higher order local Dirichlet integrals and de Branges-Rovnyak spaces

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 D‾, then such operators are of the form T=U⊕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 H(B). In fact, the space H(B) is defined in terms of a rational operator-valued Schur function B. In the case when dim⁡ker⁡T⁎=1, then H(B) can be taken to be a space of scalar-valued analytic functions in D, and the function B has a mate a defined by |B|2+|a|2=1 a.e. on ∂D. We show the mate a of a rational B is of the form a(z)=a(0)p(z)q(z), where p and q are appropriately derived from the characteristic polynomials of two associated operators. If T is a 2m-isometric expansive operator, then all zeros of p lie in the unit circle, and we completely describe the spaces H(B) by use of what we call the local Dirichlet integral of order m at the point w∈∂D.

Inequalities for the Berezin number of operators and related questions

For a bounded linear operator, acting in the reproducing kernel Hilbert space $${\mathcal {H}}={\mathcal {H}}\left( \Omega \right) $$ over some set $$\Omega $$ , its Berezin symbol (or Berezin transform) $$\widetilde{\text { }A}$$ is defined by $$\begin{aligned} {\widetilde{A}}\left( \lambda \right) :=\left\langle A{\widehat{k}}_{\lambda },{\widehat{k}}_{\lambda }\right\rangle ,\text { }\lambda \in \Omega , \end{aligned}$$ which is a bounded complex-valued function on $$\Omega ;$$ here $${\widehat{k}}_{\lambda }:=\frac{{\widehat{k}}_{\lambda }}{\left\| k_{\lambda }\right\| _{{\mathcal {H}}}}$$ is the normalized reproducing kernel of $${\mathcal {H}}$$ . The Berezin set and the Berezin number of an operator A are defined respectively by $$\begin{aligned} \mathrm {Ber}\left( A\right) :=\mathrm {Range}\left( {\widetilde{A}}\right) =\left\{ {\widetilde{A}}\left( \lambda \right) :\lambda \in \Omega \right\} \end{aligned}$$ and $$\begin{aligned} \mathrm {ber}\left( A\right) :=\sup \left\{ \left| \gamma \right| :\gamma \in \mathrm {Ber}\left( A\right) \right\} =\sup _{\lambda \in \Omega }\left| {\widetilde{A}}\left( \lambda \right) \right| . \end{aligned}$$ Since $$\mathrm {Ber}\left( A\right) \subset W\left( A\right) $$ (numerical range) and $$\mathrm {ber}\left( A\right) \le w\left( A\right) $$ (numerical radius), it is natural to investigate these new numerical quantities of operators and to get some similar results as for numerical range and numerical radius. In this paper, we prove many different type inequalities, including power inequality $$\mathrm {ber}\left( A^{n}\right) \le \mathrm {ber}\left( A\right) ^{n}$$ for the Berezin number of operators. We also study the uncertainty principle for Berezin symbols and we describe spectrum and compactness of functions of model operator in terms of Berezin symbols. Some related problems for de Branges-Rovnyak space operators are also discussed.

On kernels of Toeplitz operators

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 a result on the structure of de Branges–Rovnyak spaces generated by nonextreme functions.