De Branges Rovnyak Spaces Research Articles

Integral representation of the n-th derivative in de Branges-Rovnyak spaces and the norm convergence of its reproducing kernel

In this paper, we give an integral representation for the boundary values of derivatives of functions of the de Branges–Rovnyak spaces ℋ(b), where b is in the unit ball of H ∞ (ℂ + ). 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 evaluation of the n-th derivative of elements of ℋ(b) at the point ω as it tends radially to a point of the real axis.

Carathéodory-Julia type theorems for operator valued Schur functions

We extend the Carathéodory-Julia theorem on angular derivatives as well as its higher order analogue established recently in [4] to the setting of contractive valued functions analytic on the unit disk. Carathéodory-Julia type conditions for an operator valued Schur-class function w are shown to be equivalent to the requirement that every function from the de Branges-Rovnyak space associated with w has certain directional boundary angular derivatives.

Boundary Behavior of Functions in the de Branges–Rovnyak Spaces

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.

Reproducing kernels, de Branges-Rovnyak spaces, and norms of weighted composition operators

We prove that the norm of a weighted composition operator on the Hardy space H 2 of the disk is controlled by the norm of the weight function in the de Branges-Rovnyak space as- sociated to the symbol of the composition operator. As a corollary we obtain a new proof of the boundedness of composition operators on H 2 , and recover the standard upper bound for the norm. Sim- ilar arguments apply to weighted Bergman spaces. We also show that the positivity of a generalized de Branges-Rovnyak kernel is sufficient for the boundedness of a given composition operator on the standard functions spaces on the unit ball.

Évaluation ponctuelle et espace de Hardy : le cas multi-échelle

We define a point evaluation for transfer operators of multiscale causal dissipative systems. We associate to such a system a de Branges Rovnyak space, which serves as the state space of a coisometric realization. To cite this article: D. Alpay et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Some finite-dimensional backward-shift-invariant subspaces in the ball and a related interpolation problem

We solve Gleason's problem in the reproducing kernel Hilbert space with repoducing kernel\(1/\left( {1 - \sum\nolimits_1^N {z_j w_j^* } } \right)\). We define and study some finite-dimensional resolvent-invariant subspaces that generalize the finite-dimensional de Branges-Rovnyak spaces to the setting of the ball.

Backward Shift Invariant Operator Ranges

Results on first order Ext groups for Hilbert modules over the disk algebra are used to study certain backward shift invariant operator ranges, namely de Branges–Rovnyak spaces and a more general class called H(W;B) spaces. Necessary and sufficient conditions are given for the groups Ext1A(D)(H2C, H(W;B)) to vanish whereH2Cis thedualof the vector-valued Hardy module, H2C. One condition involves an extension problem for the Hankel operator with symbolB,ΓB, but viewed as a module map from H2Cinto H(W;B). The group Ext1A(D)(H2C, H(W;B))=(0) precisely whenΓBextends to a module map from L2Cinto H(W;B) and this in turn is equivalent to the injectivity of H(W;B) in the category of contractive HilbertA(D)-modules. This result applied to the de Branges–Rovnyak spaces yields a connection between the extension problem for the HankelΓB and the operator corona problem.

Closed commutants of the backward shift operator

Let F be a function in the Hardy space of the unit disk H. We can define the unbounded Toeplitz operator TF operating on a suitable linear submanifold of H (for instance H∞). In recent years many questions have been raised about the behavior of these operators. Most of these problems appear naturally when studying the algebra of multipliers or the backward shift invariant subspaces of the so-called de Branges-Rovnyak spaces (see [11] and [17]). If S∗ denotes the backward shift operator on H, it is not difficult to see that TF commutes with S ∗. More generally, if Q is a closed operator on some linear submanifold of H that commutes with S∗, then the domain of Q, D(Q) is dense in some (closed) S∗ invariant subspace of H. Therefore Beurling’s theorem assures that D(Q) is dense in H or in (uH2)⊥ = H(u), for some inner function u. If Q is a bounded operator on H that commutes with S∗, it is easy to see that Q = Tφ with φ ∈ H∞. The analogous result for bounded operators on H(u) that commute with S∗ u = S∗/H(u) is a well known theorem of Sarason [14]. Moreover, we can choose φ ∈ H∞ so that Q = Tφ/H(u) and ‖φ‖∞ = ‖Q‖. There are two natural questions appearing at this point. What are the closed operators that commute with S∗ (or with S∗ u)? Do the above results for bounded operators have analogous versions for closed operators? The purpose of this paper is to answer these questions in both cases, when D(Q) is dense in H and in H(u), for some inner function u. In particular, we find necessary and sufficient conditions for such an operator Q to have the form TF (or TF/H(u)) with F ∈ H. As a byproduct of this result we also obtain a short proof of Sarason’s theorem.

Local Dirichlet spaces as de Branges-Rovnyak spaces

The harmonically weighted Dirichlet spaces associated with unit point masses are shown to coincide with de Branges-Rovnyak spaces, with equality of norms. As a consequence it is shown that radial expansion operators act contractively in harmonically weighted Dirichlet spaces.

Lifting Theorem as a Special Case of Abstract Interpolation Problem

Using properties of the de Branges-Rovnyak spaces we include the classical lifting problem into the general scheme of the abstract interpolation problem.

Commuting hyponormal operators

Introduction* Let T be a completely nonunitary contraction on Hubert space such that 1 — T*T has closed range. There exists a power series B(z) = ΣBnzn with operator coefficients which converges and is bounded by one in the unit disk such that T is unitarily equivalent to the difference-quotient transformation in the de Branges-Rovnyak space 3f(B) [1, Theorem 4]. The characteristic function B{z) is said to be of scalar type if {Bn: n ^> 0} is a commuting family of normal operators. Inner functions of scalar type were introduced and characterized in [10]. In this paper, it is shown that if {Bn: n = 0, — -,N} is a commuting family of normal operators, then polynomials p(Γ) in Γ of degree at most ΛΓ (weak limits of polynomials in T if B(z) is of scalar type) which satisfy \\p(T)f ^ ||p(T)*/H for every / in the range of 1 — T*T are restrictions of