De Branges spaces and Krein's theory of entire operators

This work presents a contemporary treatment of Krein's entire operators with deficiency indices $(1,1)$ and de Branges' Hilbert spaces of entire functions. Each of these theories played a central role in the research of both renown mathematicians. Remarkably, entire operators and de Branges spaces are intimately connected and the interplay between them has had an impact in both spectral theory and the theory of functions. This work exhibits the interrelation between Krein's and de Branges' theories by means of a functional model and discusses recent developments, giving illustrations of the main objects and applications to the spectral theory of difference and differential operators.Comment: 37 pages, no figures. The abstract was extended. Typographical errors were corrected. The bibliography style was change

A restricted shift completeness problem

We solve a problem about the orthogonal complement of the space spanned by restricted shifts of functions in $L^2[0,1]$ posed by M.Carlsson and C.Sundberg.Comment: 7 page

Sur les "Espaces de Sonine" associes par de Branges a la transformation de Fourier

Nous avons obtenu des formules explicites representant les fonctions E(z) apparaissant dans la theorie des ``Espaces de Sonine'' associes par de Branges a la transformation de Fourier.Comment: 7 page

Orthogonal Newton polynomials

AbstractThe problem is to determine all nonnegative measures on the Borel subsets of the complex plane with respect to which all polynomials are square integrable and with respect to which the Newton polynomials form an orthogonal set. The Newton polynomials do not belong to any classical scheme of orthogonal polynomials. The discovery that a plane measure exists with respect to which they form an orthogonal set was only recently made by T. L. Kriete and D. Trutt [Amer. J. Math.93 (1971), 215–225]. A general structure theory for such measures is now obtained under hypotheses suggested by the expansion theory of Cesàro operators

An adelic causality problem related to abelian L-functions

I associate to a global field K a Lax-Phillips scattering which has the property of causality if and only if the Riemann Hypothesis holds for all the abelian L-functions of K. As a Hilbert space closure problem this provides an adelic variation on a theme initiated by Nyman and Beurling. The adelic aspects are related to previous work by Tate, Iwasawa and Connes.Comment: 18 pages, latex2e with amsfonts. Final version, accepted for publicatio

de Branges-Rovnyak spaces: basics and theory

For $S$ a contractive analytic operator-valued function on the unit disk ${\mathbb D}$, de Branges and Rovnyak associate a Hilbert space of analytic functions ${\mathcal H}(S)$ and related extension space ${\mathcal D(S)}$ consisting of pairs of analytic functions on the unit disk ${\mathbb D}$. This survey describes three equivalent formulations (the original geometric de Branges-Rovnyak definition, the Toeplitz operator characterization, and the characterization as a reproducing kernel Hilbert space) of the de Branges-Rovnyak space ${\mathcal H}(S)$, as well as its role as the underlying Hilbert space for the modeling of completely non-isometric Hilbert-space contraction operators. Also examined is the extension of these ideas to handle the modeling of the more general class of completely nonunitary contraction operators, where the more general two-component de Branges-Rovnyak model space ${\mathcal D}(S)$ and associated overlapping spaces play key roles. Connections with other function theory problems and applications are also discussed. More recent applications to a variety of subsequent applications are given in a companion survey article

Sur certains espaces de Hilbert de fonctions entieres, lies a la transformation de Fourier et aux fonctions L de Dirichlet et de Riemann

We construct in a Sonine Space of entire functions a subspace related to the Riemann zeta function and we show that the quotient contains vectors intrinsically attached to the non-trivial zeros and their multiplicities.Comment: 10 pages. In french with an english summar

Des equations de Dirac et de Schrodinger pour la transformation de Fourier

Dyson a associe aux determinants de Fredholm des noyaux de Dirichlet pairs (resp. impairs) une equation de Schrodinger sur un demi-axe et a employe les methodes du scattering inverse de Gel'fand-Levitan et de Marchenko, en tandem, pour etudier l'asymptotique de ces determinants. Nous avons propose suite a notre mise-au-jour de l'operateur conducteur de chercher a realiser la transformation de Fourier elle-meme comme un scattering, et nous obtenons ici dans ce but deux systemes de Dirac sur l'axe reel tout entier et qui sont associes intrinsequement, respectivement, aux transformations en cosinus et en sinus. (Dyson has associated with the Fredholm determinants of the even (resp. odd) Dirichlet kernels a Schrodinger equation on the half-axis and has used, in tandem, the Gel'fand-Levitan and Marchenko methods of inverse scattering theory to study the asymptotics of these determinants. We have proposed following our unearthing of the conductor operator to seek to realize the Fourier transform itself as a scattering, and we obtain here to this end two Dirac systems on the entire real axis which are intrinsically associated, respectively, to the cosine and to the sine transforms.)Comment: 8 pages, with a summary in English. One or two things adde