Universality limits for random matrices and de Branges spaces of entire functions

We prove that de Branges spaces of entire functions describe universality limits in the bulk for random matrices, in the unitary case. In particular, under mild conditions on a measure with compact support, we show that each possible universality limit is the reproducing kernel of a de Branges space of entire functions that equals a classical Paley-Wiener space. We also show that any such reproducing kernel, suitably dilated, may arise as a universality limit for sequences of measures on [−1,1].     

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 L² (\mathbbR, dn){L^2 (\mathbb{R}, d\nu)} onto a de Branges space of entire functions which acts as multiplication by a measurable function.     

De Branges Spaces of Entire Functions Symmetric About the Origin

We define and investigate the class of symmetric and the class of semibounded de Branges spaces of entire functions. A construction is made which assigns to each symmetric de Branges space a semibounded one. By employing operator theoretic tools it is shown that every semibounded de Branges space can be obtained in this way, and which symmetric spaces give rise to the same semibounded space. Those subclasses of Hermite-Biehler functions are determined which correspond to symmetric or semibounded, respectively, nondegenerated de Branges spaces. The above assignment is determined in terms of the respective generating Hermite-Biehler functions.     

Majorization in de Branges spaces I. Representability of subspaces

In this series of papers we study subspaces of de Branges spaces of entire functions which are generated by majorization on subsets D of the closed upper half-plane. The present, first, part is addressed to the question which subspaces of a given de Branges space can be represented by means of majorization. Results depend on the set D where majorization is permitted. Significantly different situations are encountered when D is close to the real axis or accumulates to i∞.     

On the nature of the de Branges Hamiltonian

We prove the theorem announced by the author in 1995 in the paper “A criterion for the discreteness of the spectrum of a singular canonical system” (Funkts. Anal. Prilozhen., 29, No. 3).In developing the theory of Hilbert spaces of entire functions (we call them Krein-de Branges spaces), de Branges arrived at a certain class of canonical equations of phase dimension 2. He showed that, for any given Krein-de Branges space, there exists a canonical equation of the class indicated that restores a chain of Krein-de Branges spaces imbedded one into another. The Hamiltonians of such canonical equations are called de Branges Hamiltonians. The following question arises: Under what conditions will the Hamiltonian of a certain canonical equation be a de Branges Hamiltonian? The main theorem of the present work, together with Theorem 1 of the paper cited above, gives an answer to this question.     

De Branges spaces and Fock spaces

Abstract

Relations between two classes of Hilbert spaces of entire functions, de Branges spaces and Fock-type spaces with nonradial weights, are studied. It is shown that any de Branges space can be realized as a Fock-type space with equivalent area norm, and several constructions of a representing weight are suggested. For some special classes of weights (e.g. weights depending on the imaginary part only) the corresponding de Branges spaces are explicitly described.     

Existence of zerofree functions N-associated to a de Branges Pontryagin space

In the theory of de Branges Hilbert spaces of entire functions, so-called ‘functions associated to a space’ play an important role. In the present paper we deal with a generalization of this notion in two directions, namely with functions N-associated (N ? \mathbb Z)({N \in\mathbb {Z}}) to a de Branges Pontryagin space. Let a de Branges Pontryagin space P{\mathcal {P}} and N ? \mathbb Z{N \in \mathbb {Z}} be given. Our aim is to characterize whether there exists a real and zerofree function N-associated to P{\mathcal {P}} in terms of Kreĭn’s Q-function associated with the multiplication operator in P{\mathcal {P}} . The conditions which appear in this characterization involve the asymptotic distribution of the poles of the Q-function plus a summability condition. Although this question may seem rather abstract, its answer has a variety of nontrivial consequences. We use it to answer two questions arising in the theory of general (indefinite) canonical systems. Namely, to characterize whether a given generalized Nevanlinna function is the intermediate Weyl-coefficient of some system in terms of its poles and residues, and to characterize whether a given general Hamiltonian ends with a specified number of indivisible intervals in terms of the Weyl-coefficient associated to the system. In addition, we present some applications, e.g., dealing with admissible majorants in de Branges spaces or the continuation problem for hermitian indefinite functions.     

Existence of zerofree functions <Emphasis Type="Italic">N</Emphasis>-associated to a de Branges Pontryagin space

In the theory of de Branges Hilbert spaces of entire functions, so-called ‘functions associated to a space’ play an important role. In the present paper we deal with a generalization of this notion in two directions, namely with functions N-associated \(({N \in\mathbb {Z}})\) to a de Branges Pontryagin space. Let a de Branges Pontryagin space \({\mathcal {P}}\) and \({N \in \mathbb {Z}}\) be given. Our aim is to characterize whether there exists a real and zerofree function N-associated to \({\mathcal {P}}\) in terms of Kre?n’s Q-function associated with the multiplication operator in \({\mathcal {P}}\) . The conditions which appear in this characterization involve the asymptotic distribution of the poles of the Q-function plus a summability condition. Although this question may seem rather abstract, its answer has a variety of nontrivial consequences. We use it to answer two questions arising in the theory of general (indefinite) canonical systems. Namely, to characterize whether a given generalized Nevanlinna function is the intermediate Weyl-coefficient of some system in terms of its poles and residues, and to characterize whether a given general Hamiltonian ends with a specified number of indivisible intervals in terms of the Weyl-coefficient associated to the system. In addition, we present some applications, e.g., dealing with admissible majorants in de Branges spaces or the continuation problem for hermitian indefinite functions.     

Pontryagin spaces of entire functions I

We give a generalization of L.de Branges theory of Hilbert spaces of entire functions to the Pontryagin space setting. The aim of this-first-part is to provide some basic results and to investigate subspaces of Pontryagin spaces of entire functions. Our method makes strong use of L.de Branges's results and of the extension theory of symmetric operators as developed by M.G.Krein.     

Hermitian indefinite functions and Pontryagin spaces of entire functions

We establish and investigate a connection between hermitian indefinite continuous functions with finitely many negative squares defined on a finite interval and so-called de Branges spaces of entire functions. This enables us to relate to any hermitian indefinite continuous function on the real axis a certain chain of 2×2-matrix valued entire functions, which are in the positive definite case tightly connected with canonical systems of differential equations.     

Shifted Hermite-Biehler Functions and Their Applications

We investigate a particular subclass of so-called symmetric indefinite Hermite-Biehler functions and give a characterization of functions of this class in terms of the location of their zeros. For the proof we employ the theory of de Branges Pontryagin spaces of entire functions. We apply our results to obtain information on the eigenvalues of some boundary value problems.     

Pontryagin spaces of entire functions II

We continue the study of a generalization of L. de Branges's theory of Hilbert spaces of entire functions to the Pontryagin space setting. In this-second-part we investigate isometric embeddings of spaces of entire functions into spacesL ² () understood in a distributional sense and consider Weyl coefficients of matrix chains. The main task is to give a proof of an indefinite version of the inverse spectral theorem for Nevanlinna functions. Our methods use the theory developed by L. de Branges and the theory of extensions of symmetric operators of M.G.Krein.     

Classification theorems for operators preserving zeros in a strip

We characterize all linear operators which preserve certain spaces of entire functions whose zeros lie in a closed strip. Necessary and sufficient conditions are obtained for the related problem with real entire functions, and some classical theorems of de Bruijn and Pólya are extended. Specifically, we reveal new differential operators which map real entire functions whose zeros lie in a strip to real entire functions whose zeros lie in a narrower strip; this is one of the properties that characterize a “strong universal factor” as defined by de Bruijn. Using elementary methods, we prove a theorem of de Bruijn and extend a theorem of de Bruijn and Ilieff which states a sufficient condition for a function to have a Fourier transform with only real zeros.     

A dynamical system approach to Heisenberg uniqueness pairs

A canonical system is a kind of first-order system of ordinary differential equations on an interval of the real line parametrized by complex numbers. It is known that any solution of a canonical system generates an entire function of the Hermite-Biehler class. In this paper, we deal with the inverse problem to recover a canonical system from a given entire function of the Hermite-Biehler class satisfying appropriate conditions. This inverse problem was solved by de Branges in 1960s. However his results are often not enough to investigate a Hamiltonian of recovered canonical system. In this paper, we present an explicit way to recover a Hamiltonian from a given exponential polynomial belonging to the Hermite-Biehler class. After that, we apply it to study distributions of roots of self-reciprocal polynomials.     

Finite Dimensional de Branges Spaces on Riemann Surfaces

We study certain finite dimensional reproducing kernel indefinite inner product spaces of multiplicative half order differentials on a compact real Riemann surface; these spaces are analogues of the spaces introduced by L. de Branges when the Riemann sphere is replaced by a compact real Riemann surface of a higher genus. In de Branges theory an important role is played by resolvent-like difference quotient operators Rα; here we introduce generalized difference quotient operators Ryα for any non-constant meromorphic function y on the Riemann surface. The spaces we study are invariant under generalized difference quotient operators and can be characterized as finite dimensional indefinite inner product spaces invariant under two operators Ry1αi and Ry2α2, where y1 and y2 generate the field of meromorphic functions on the Riemann surface, which satisfy a supplementary identity, analogous to the de Branges identity for difference quotients. Just as the classical de Branges spaces and difference quotient operators appear in the operator model theory for a single nonselfadjoint (or nonunitary) operator, the spaces we consider and generalized difference quotient operators appear in the model theory for commuting nonselfadjoint operators with finite nonhermitian ranks.     

Polynomials in the de Branges spaces of entire functions

We study the problem of density of polynomials in the de Branges spaces ℋ(E) of entire functions and obtain conditions (in terms of the distribution of the zeros of the generating function E) ensuring that the polynomials belong to the space ℋ(E) or are dense in this space. We discuss the relation of these results with the recent paper of V. P. Havin and J. Mashreghi on majorants for the shift-coinvariant subspaces. Also, it is shown that the density of polynomials implies the hypercyclicity of translation operators in ℋ(E).     

Uniform boundedness of the derivatives of meromorphic inner functions on the real line

Inner functions are an important and popular object of study in the field of complex function theory. We look at meromorphic inner functions with a given spectrum and provide sufficient conditions for them to have uniformly bounded derivative on the real line. This question was first studied by Louis de Branges in 1968 and was later revived by Anton Baranov in 2011.     

Which perturbations of quasianalytic weights preserve quasianalyticity? How to use de Branges’ theorem

Based on an approach of de Branges and the theory of entire functions, we prove two results pertaining to the Bernstein approximation problem, one concerning analytic perturbations of quasianalytic weights and the other dealing with density of polynomials in spaces with nonsymmetric weights. We improve earlier results of V. P. Gurarii and A. Volberg, giving a more complete answer to a question posed by L. Ehrenpreis and S. N. Mergelyan.