De Branges Rovnyak Spaces Research Articles

Gleason’s problem associated to the fractional Cauchy-Riemann operator, Fueter series, Drury-Arveson space and related topics

In this paper we present the building blocks for a function theory based on fractional Cauchy-Riemann operators. We are going to construct basic monogenic powers and Fueter series. With these tools we are going to study Gleason’s problem and reproducing kernel spaces, like the Drury-Arveson space and de Branges-Rovnyak spaces. We end with a statement on Schur multipliers in this setting.

Partial orders on partial isometries

This paper studies three natural pre-orders of increasing generality on the set of all completely non-unitary partial isometries with equal defect indices. We show that the problem of determining when one partial isometry is less than another with respect to these pre-orders is equivalent to the existence of a bounded (or isometric) multiplier between two natural reproducing kernel Hilbert spaces of analytic functions. For large classes of partial isometries these spaces can be realized as the well-known model subspaces and deBrangesRovnyak spaces. This characterization is applied to investigate properties of these pre-orders and the equivalence classes they generate.

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.

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.

Weighted sub-Bergman Hilbert spaces

AbstractWe consider Hilbert spaces which are counterparts of the de Branges-Rovnyak spaces in the context of the weighted Bergman spaces A

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.

Schur Functions and Their Realizations in the Slice Hyperholomorphic Setting

In this paper we start the study of Schur analysis in the quaternionic setting using the theory of slice hyperholomorphic functions. The novelty of our approach is that slice hyperholomorphic functions allow to write realizations in terms of a suitable resolvent, the so called S-resolvent operator and to extend several results that hold in the complex case to the quaternionic case. We discuss reproducing kernels and positive definite functions in this setting and we show how they can be obtained using the extension operator and the slice hyperholomorphic product. We define Schur multipliers, and find their co-isometric realization in terms of the associated de Branges–Rovnyak space.

Abstract Interpolation in Vector-Valued de Branges–Rovnyak Spaces

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 also paid to the case that the map which provides this description is injective. The interpolation problem studied here contains as particular cases (1) the vector-valued version of the interpolation problem with operator argument considered recently in Ball et al. (Proc Am Math Soc 139(2), 609-618, 2011) (for the nondegenerate and scalar-valued case) and (2) a boundary interpolation problem in $\mathcal{H}(K_S)$. In addition, we discuss connections with results on kernels of Toeplitz operators and nearly invariant subspaces of the backward shift operator.

De Branges–Rovnyak spaces and Dirichlet spaces

Sarason has shown that the local Dirichlet spaces D λ may be considered as manifestations of de Branges–Rovnyak spaces H ( b ) , and has used this identification to give a new proof that the spaces D λ are star-shaped. We investigate which other Dirichlet spaces D ( μ ) arise as de Branges–Rovnyak spaces, and which other de Branges–Rovnyak spaces H ( b ) are star-shaped. We also prove a transfer principle which represents H ( b ) -spaces inside D λ .

Interpolation in de Branges-Rovnyak spaces

A general interpolation problem with operator argument is studied for functions from the de Branges-Rovnyak space associated with an analytic function mapping the open unit disk into the closed unit disk. The interpolation condition is taken in the Rosenblum-Rovnyak form (with a suitable interpretation of ) for given Hilbert space operator and two vectors from the same space.

Weighted norm inequalities for de Branges-Rovnyak spaces and their applications

Let ${\cal H}(b)$ denote the de Branges--Rovnyak space associated with a function $b$ in the unit ball of $H^\infty({\Bbb C}_+)$. We study the boundary behavior of the derivatives of functions in ${\cal H}(b)$ and obtain weighted norm estimates of the form $\|f^{(n)}\|_{L^2(\mu)} \le C\|f\|_{{\cal H}(b)}$, where $f \in {\cal H}(b)$ and $\mu$ is a Carleson-type measure on ${\Bbb C}_+\cup{\Bbb R}$. We provide several applications of these inequalities. We apply them to obtain embedding theorems for ${\cal H}(b)$ spaces. These results extend Cohn and Volberg--Treil embedding theorems for the model (star-invariant) subspaces which are special classes of de Branges--Rovnyak spaces. We also exploit the inequalities for the derivatives to study stability of Riesz bases of reproducing kernels $\{k^b_{\lambda_n}\}$ in ${\cal H}(b)$ under small perturbations of the points $\lambda_n$.

Mixed Invariant Subspaces over the Bidisk

It is known that the structure of invariant subspaces I of the Hardy space H2 over the bidisk is extremely complicated. One reason is that it is difficult to describe infinite dimensional wandering spaces \({I\ominus zI}\) completely. In this paper, we study the structure of nontrivial closed subspaces N of H2 with \({T_zN\subset N}\) and \({T^*_wN\subset N}\), which are called mixed invariant subspaces under Tz and \({T^*_w}\). We know that the dimension of \({N\ominus zN}\) ranges from 1 to ∞. If \({T^*_w(N\ominus zN)\subset N\ominus zN}\), we may describe N completely. If \({T^*_w(N\ominus zN)\not\subset N\ominus zN}\), it seems difficult to describe N generally. So we study N under the condition \({dim\,(N\ominus zN)=1}\). Write \({M=H^2\ominus N}\). We describe \({M\ominus wM}\) precisely. We give a characterization of N for which there is a nonzero function \({\varphi}\) in \({M\ominus wM}\) satisfying \({z^k\varphi\in M\ominus wM}\) for every k ≥ 0. We also see that the space \({M\ominus wM}\) has a deep connection with the de Branges–Rovnyak spaces studied by Sarason.

Generalized Schur Functions and Related de Branges-Rovnyak Spaces in the Banach Space Setting

We study the analogues of de Branges-Rovnyak spaces in the Banach space case. An important role is played by self-adjoint operators from the dual of a Banach space into the Banach space itself. A factorization theorem for such operators is proved in the case when they have a finite number of negative squares.

Representation of Simple Symmetric Operators with Deficiency Indices (1, 1) in de Branges Space

Recently it has been shown that any regular simple symmetric operator with deficiency indices (1, 1) is unitarily equivalent to the operator of multiplication in a reproducing kernel Hilbert space of functions on the real line with the Kramer sampling property. This work has been motivated, in part, by potential applications to signal processing and mathematical physics. In this paper we exploit well-known results about de Branges–Rovnyak spaces and characteristic functions of symmetric operators to prove that any such a symmetric operator is in fact unitarily equivalent to multiplication by the independent variable in a de Branges space of entire functions. This leads to simple new results on the spectra of such symmetric operators, on when multiplication by z is densely defined in de Branges–Rovnyak spaces in the upper half plane, and to sufficient conditions for there to be an isometry from a given subspace of \({L^2 (\mathbb{R}, d\nu)}\) onto a de Branges space of entire functions which acts as multiplication by a measurable function.

On certain Riesz families in vector-valued de Branges–Rovnyak spaces

We obtain criteria for the Riesz basis property for families of reproducing kernels in vector-valued de Branges–Rovnyak spaces H(b). In particular, it is shown that in several situations the property implies a special form for the function b. We also study the completeness of a related family.

A Characterization of Schur Multipliers Between Character-Automorphic Hardy Spaces

We give a new characterization of character-automorphic Hardy spaces of order 2 and of their contractive multipliers in terms of de Branges Rovnyak spaces. Keys tools in our arguments are analytic extension and a factorization result for matrix-valued analytic functions due to Leech.