Density of disk algebra functions in de Branges–Rovnyak spaces

Alexandru Aleman; Bartosz Malman

doi:10.1016/j.crma.2017.07.015

Complex analysis/Functional analysis

Density of disk algebra functions in de Branges–Rovnyak spaces

Presented by: the Editorial Board

Alexandru Aleman ¹; Bartosz Malman ¹

¹ Centre for Mathematical Sciences, Lund University, P.O Box 118, SE-22100 Lund, Sweden

Comptes Rendus. Mathématique, Volume 355 (2017) no. 8, pp. 871-875.

Abstract (OV)
Résumé

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 $H^{\infty}$ . Together with previous theorems, it follows that this class of functions is dense in any de Branges–Rovnyak space.

On démontre que les fonctions analytiques dans le disque unité et continues dans le disque fermé sont denses dans l'espace de Branges–Rovnyak généré par un point extrémal de la boule unité de $H^{\infty}$ . En utilisant aussi des théorèmes précédents, il résulte que cette classe de fonctions est dense dans un espace de Branges–Rovnyak quelconque.

Received: 2017-04-21
Accepted: 2017-07-20
Published online: 2017-08-01

DOI: 10.1016/j.crma.2017.07.015

Author's affiliations:

Alexandru Aleman ¹; Bartosz Malman ¹

¹ Centre for Mathematical Sciences, Lund University, P.O Box 118, SE-22100 Lund, Sweden

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     author = {Alexandru Aleman and Bartosz Malman},
     title = {Density of disk algebra functions in de {Branges{\textendash}Rovnyak} spaces},
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     language = {en},
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Alexandru Aleman; Bartosz Malman. Density of disk algebra functions in de Branges–Rovnyak spaces. Comptes Rendus. Mathématique, Volume 355 (2017) no. 8, pp. 871-875. doi : 10.1016/j.crma.2017.07.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.07.015/

References
Cited by

[1] A.B. Aleksandrov Invariant subspaces of shift operators. An axiomatic approach, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), Volume 113 (1981), pp. 7-26 (264, English translation in J. Sov. Math., 22, 1983, pp. 1695-1708)

[2] A. Aleman; N.S. Feldman; W.T. Ross The Hardy space of a slit domain, Frontiers in Mathematics, Birkhäuser Verlag, Basel, Switzerland, 2009

[3] A. Aleman; S. Richter; C. Sundberg Beurling's theorem for the Bergman space, Acta Math., Volume 177 (1996), pp. 275-310

[4] S. Banach Theory of Linear Operations, N.-Holl. Math. Libr., vol. 38, North-Holland Publishing Co., Amsterdam, 1987

[5] C. Bénéteau; A. Condori; C. Liaw; W.T. Ross; A. Sola Some open problems in complex and harmonic analysis: report on problem session held during the conference Completeness problems, Carleson measures, and spaces of analytic functions, Recent Progress on Operator Theory and Approximation in Spaces of Analytic Functions, Contemp. Math., vol. 679, American Mathematical Society, Providence, RI, USA, 2016, pp. 207-217

[6] J. Cima; A. Matheson; W.T. Ross The Cauchy Transform, Math. Surv. Monogr., vol. 125, American Mathematical Society, Providence, RI, USA, 2006

[7] O. El-Fallah; E. Fricain; K. Kellay; J. Mashreghi; T. Ransford Constructive approximation in de Branges–Rovnyak spaces, Constr. Approx., Volume 44 (2016), pp. 269-281

[8] V.P. Havin; B. Jöricke The Uncertainty Principle in Harmonic Analysis, Encyclopaedia Math. Sci., vol. 72, Springer, Berlin, 1995

[9] D. Sarason Sub-Hardy Hilbert Spaces in the Unit Disk, Univ. Arkansas Lect. Notes Math. Sci., vol. 10, John Wiley & Sons, Inc., New York, 1994

[10] S.A. Vinogradov Properties of multipliers of integrals of Cauchy–Stieltjes type, and some problems of factorization of analytic functions, Drogobych, 1974, Volume 115 (1976), pp. 5-39 (in Russian); English translation in Transl. Amer. Math. Soc. (2), 1980, pp. 1-32

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