De Branges Rovnyak Research Articles

Composition Operators on de Branges–Rovnyak Spaces

We study the compactness of the composition operator on de Branges–Rovnyak spaces. Inspired by a paper by Lyubarskii–Malinnikova on model spaces, we give some necessary and some sufficient conditions for compactness. In the paper of Lyubarskii-Malinnikova, the key point is some Bernstein inequality on model spaces due to Cohn (and based on a deep inequality of Axler–Chang–Sarason involving the Hardy–Littlewood maximal function). We generalize the result of Cohn to some subspace of a de Branges–Rovnyak space (in many cases dense) and then get a sufficient condition (analogue to Lyubarskii–Malinnikova’s condition) for compactness of the composition operator on that subspace.

Examples of de Branges–Rovnyak spaces generated by nonextreme functions

Examples of de Branges–Rovnyak spaces generated by nonextreme functions

Range Spaces of Co-Analytic Toeplitz Operators

AbstractIn this paper we discuss the range of a co-analytic Toeplitz operator. These range spaces are closely related to de Branges–Rovnyak spaces (in some cases they are equal as sets). In order to understand its structure, we explore when the range space decomposes into the range of an associated analytic Toeplitz operator and an identifiable orthogonal complement. For certain cases, we compute this orthogonal complement in terms of the kernel of a certain Toeplitz operator on the Hardy space, where we focus on when this kernel is a model space (backward shift invariant subspace). In the spirit of Ahern–Clark, we also discuss the non-tangential boundary behavior in these range spaces. These results give us further insight into the description of the range of a co-analytic Toeplitz operator as well as its orthogonal decomposition. Our Ahern–Clark type results, which are stated in a general abstract setting, will also have applications to related sub-Hardy Hilbert spaces of analytic functions such as the de Branges–Rovnyak spaces and the harmonically weighted Dirichlet spaces.

Hilbert spaces contractively contained in weighted Bergman spaces on the unit disk

Sub-Bergman Hilbert spaces are analogues of de Branges–Rovnyak spaces in the Bergman space setting. They are reproducing kernel Hilbert spaces contractively contained in the Bergman space of the unit disk. K. Zhu analyzed sub-Bergman Hilbert spaces associated with finite Blaschke products, and proved that they are norm equivalent to the Hardy space. Later S. Sultanic found a different proof of Zhu's result, which works in weighted Bergman space settings as well. In this paper, we give a new approach to this problem and obtain a stronger result. Our method relies on the theory of reproducing kernel Hilbert spaces.

Hilbert spaces of analytic functions with a contractive backward shift

We consider Hilbert spaces of analytic functions in the disk with a normalized reproducing kernel and such that the backward shift f(z)↦f(z)−f(0)z is a contraction on the space. We present a model for this operator and use it to prove the surprising result that functions which extend continuously to the closure of the disk are dense in the space. This has several applications, for example we can answer a question regarding reverse Carleson embeddings for these spaces. We also identify a large class of spaces which are similar to the de Branges–Rovnyak spaces and prove some results which are new even in the classical case.

Density of polynomials in sub-Bergman Hilbert spaces

The sub-Bergman Hilbert spaces are analogues of de Branges–Rovnyak spaces in the Bergman space setting. We prove that the polynomials are dense in sub-Bergman Hilbert spaces. This answers the question posted by Zhu in the affirmative.

Bounded Composition Operators and Multipliers of Some Reproducing Kernel Hilbert Spaces on the Bidisk

We study the boundedness of composition operators on the bidisk using reproducing kernels. We show that a composition operator is bounded on the Hardy space $$H^2(\mathbb {D}^2)$$ if some associated function is a positive kernel. This positivity condition naturally leads to the study of the sub-Hardy Hilbert spaces of the bidisk, which are analogs of de Branges–Rovnyak spaces on the unit disk. We discuss multipliers of those spaces and obtain some classes of bounded composition operators on the bidisk.

Aleksandrov–Clark Theory for Drury–Arveson Space

Recent work has demonstrated that Clark’s theory of unitary perturbations of the backward shift on a deBranges–Rovnyak space on the disk has a natural extension to the several-variable setting. In the several-variable case, the appropriate generalization of the Schur class of contractive analytic functions is the closed unit ball of the Drury–Arveson multiplier algebra and the Aleksandrov–Clark measures are necessarily promoted to positive linear functionals on a symmetrized subsystem of the Free Disk operator system $$\mathcal {A} _d + \mathcal {A} _d ^*$$ , where $$\mathcal {A} _d$$ is the Free or Non-commutative Disk Algebra on d generators. We continue this program for vector-valued Drury–Arveson space by establishing the existence of a canonical ‘tight’ extension of any Aleksandrov–Clark map to the full Free Disk operator system. We apply this tight extension to generalize several earlier results and we characterize all extensions of the Aleksandrov–Clark maps.

Representing systems generated by reproducing kernels

In this paper, we study the representing and absolutely representing systems in the context of reproducing kernel Hilbert spaces. We prove in particular that, in many classical spaces including weighted Hardy and Dirichlet spaces and de Branges–Rovnyak spaces, there cannot exist absolutely representing systems of reproducing kernels.

A Conservative de Branges–Rovnyak Functional Model for Operator Schur Functions on $$\mathbb C^+$$ C +

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 classical uniqueness result (which states uniqueness up to unitary similarity), we introduce non-invertible intertwinements of system nodes.

Density of disk algebra functions in de Branges–Rovnyak spaces

We prove that functions analytic in the unit disk and continuous up to the boundary are dense in the de Branges–Rovnyak spaces induced by the extreme points of the unit ball of H∞. Together with previous theorems, it follows that this class of functions is dense in any de Branges–Rovnyak space.

Clark model in the general situation

For a unitary operator U in a Hilbert space H the family of its unitary perturbations by rank one operators with fixed range is parametrized by a complex parameter γ, ǀγǀ = 1. Namely, all such unitary perturbations are operators U γ := U + (γ − 1)( ·, b 1) H b, where b ∈ H, ǁbǁ = 1, b 1 = U −1 b, ǀγǀ = 1. For ǀγǀ < 1, the operators U γ are contractions with one-dimensional defects. Restricting our attention to the non-trivial part of perturbation, we assume that b is a cyclic vector for U, i.e., $${\rm H} = \overline {span} \left\{ {{U^n}b:n \in \mathbb{Z}} \right\}$$ . In this case, the operator U γ, ǀγǀ < 1 is a completely non-unitary contraction and thus unitarily equivalent to its functional model M γ , which is the compression of the multiplication by the independent variable z onto the model space $${K_{{\theta _\gamma }}}$$ ; here, θ γ is the characteristic function of the contraction U γ . The Clark operator Φ γ is a unitary operator intertwining the operator U γ , ǀγǀ < 1 (in the spectral representation of the operator U) and its model M γ , M γ Φ γ = Φ γ U γ . In the case when the spectral measure of U is purely singular (equivalently, the characteristic function θ γ is inner), the operator Φ γ was described from a slightly different point of view by D. Clark [3]. The case where θ γ is an extreme point of the unit ball in H ∞ was treated by D. Sarason [18], using the sub-Hardy spaces H(θ) introduced by L. de Branges. In this paper, we treat the general case and give a systematic presentation of the subject. We first find a formula for the adjoint operator Φ * , which is represented by a singular integral operator, generalizing in a sense the normalized Cauchy transform studied by A. Poltoratskii [16]. We begin by presenting a “universal” representation that works for any transcription of the functional model. We then give the formulas adapted for specific transcriptions of the model, such as Sz.-Nagy–Foias and the de Branges–Rovnyak transcriptions. Finally, we obtain the representation of Φ γ .

Composition operators induced by injective homomorphisms on infinite weighted graphs

We deal with composition operators induced by injective homomorphisms on infinite weighted graphs from a viewpoint of reproducing kernel Hilbert space theory. The representation formula for their adjoint operators is given in the terms of the frames constructed from reproducing kernels, and de Branges–Rovnyak spaces induced by composition operators are studied.

Concrete Examples of $$\varvec{\mathscr {H}(b)}$$ H ( b ) Spaces

In this paper, we give an explicit description of de Branges–Rovnyak spaces \(\mathscr {H}(b)\) 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.

Dirichlet Spaces with Superharmonic Weights and de Branges–Rovnyak Spaces

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 $$\begin{aligned} \mathcal{D}_\omega (f_r)\le \frac{2r}{1+r}\mathcal{D}_\omega (f) \qquad (0\le r<1), \end{aligned}$$ where \(\mathcal{D}_\omega \) denotes the Dirichlet integral with superharmonic weight \(\omega \), and \(f_r(z):=f(rz)\) is the r-dilation of the holomorphic function f.

Two-Isometries and de Branges–Rovnyak Spaces

We characterize the symbols of the de Branges–Rovnyak spaces for which the shift operator is concave or 2-isometry. As applications, we consider wandering \(z\)-invariant subspaces and equality between a de Branges–Rovnyak space and a Dirichlet type space.

De Branges–Rovnyak Realizations of Operator-Valued Schur Functions on the Complex Right Half-Plane

We give a controllable energy-preserving and an observable co-energy-preserving de Branges-Rovnyak functional model realization of an arbitrary given operator Schur function defined on the complex right-half plane. We work the theory out fully in the right-half plane, without using results for the disk case, in order to expose the technical details of continuous-time systems theory. At the end of the article, we make explicit the connection to the corresponding classical de Branges-Rovnyak realizations for Schur functions on the complex unit disk.

Nonextreme de Branges–Rovnyak Spaces as Models for Contractions

The de Branges–Rovnyak spaces are known to provide an alternate functional model for contractions on a Hilbert space, equivalent to the Sz.-Nagy–Foias model. The scalar de Branges–Rovnyak spaces $${\mathcal{H}(b)}$$ have essentially different properties, according to whether the defining function b is or not extreme in the unit ball of H ∞. For b extreme the model space is just $${\mathcal{H}(b)}$$ , while for b nonextreme an additional construction is required. In the present paper we identify the precise class of contractions which have as a model $${\mathcal{H}(b)}$$ with b nonextreme.

Which de Branges–Rovnyak spaces are Dirichlet spaces (and vice versa)?

We investigate for which pairs b,μ the de Branges–Rovnyak space Hb is equal to the generalized Dirichlet space Dμ.

Unitary Perturbations of Compressed n-Dimensional Shifts

Given any contractive matrix-valued analytic function $${\Theta}$$ on the unit disc $${\mathbb{D}}$$ , we construct a $${\mathcal{U}(n)}$$ -parameter family of unitary operators which correspond to $${\Theta}$$ in a natural way. These operators are unitarily equivalent to higher dimensional analogues of Clark’s unitary perturbations, a family of unitary operators associated to any self-map of the unit disc. Clark’s unitary perturbations were introduced in a seminal paper of Clark which has inspired the study of what are now called Aleksandrov–Clark measures. Our higher dimensional analogues of Clark’s unitary perturbations are applied to obtain matrix-generalizations of several classical results on the Aleksandrov–Clark measures associated to any holomorphic self-map of the unit disc. In particular we establish a matrix-valued Aleksandrov disintegration theorem for the Aleksandrov–Clark measures associated with matrix-valued contractive analytic functions $${\Theta}$$ , and, by following results of Clark and Fricain in the scalar case, we provide a necessary and sufficient condition for the de Branges–Rovnyak space associated with $${\Theta}$$ to contain a total orthogonal set of point evaluation vectors.