Functional models and asymptotically orthonormal sequences
[Modèles fonctionnels et suites asymptotiquement orthonormales]
Annales de l'Institut Fourier, Tome 53 (2003) no. 5, pp. 1527-1549.

Supposons que H 2 est l’espace de Hardy du disque unité du plan complexe et Θ une fonction intérieure. On donne des conditions pour qu’une suite de noyaux reproduisants normalisés dans l’espace modèle K Θ =H 2 ΘH 2 soit asymptotiquement proche d’une suite orthonormale. La question de la complétude est aussi étudiée.

Suppose H 2 is the Hardy space of the unit disc in the complex plane, while Θ is an inner function. We give conditions for a sequence of normalized reproducing kernels in the model space K Θ =H 2 ΘH 2 to be asymptotically close to an orthonormal sequence. The completeness problem is also investigated.

DOI : 10.5802/aif.1987
Classification : 47A45, 46E22, 46B15, 30D55
Keywords: Hardy space, functional model, asymptotically orthornormal sequence
Mot clés : espace de Hardy, modèle fonctionnel, suite asymptotiquement orthonormale
Chalendar, Isabelle 1 ; Fricain, Emmanuel 1 ; Timotin, Dan 2

1 Université Claude Bernard Lyon I, Institut Girard Desargues, UFR de Mathématiques, 69622 Villeurbanne Cedex (France)
2 Institute of Mathematics of the Romanian Academy, PO Box 1-764, Bucharest 70700 (Romania)
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Chalendar, Isabelle; Fricain, Emmanuel; Timotin, Dan. Functional models and asymptotically orthonormal sequences. Annales de l'Institut Fourier, Tome 53 (2003) no. 5, pp. 1527-1549. doi : 10.5802/aif.1987. http://www.numdam.org/articles/10.5802/aif.1987/

[1] I. Boricheva Geometric properties of projections of reproducing kernels on z * -invariant subspaces of H 2 , Journal of Functional Analysis, Volume 161 (1999), pp. 397-417 | DOI | MR | Zbl

[2] D. Clark One dimensional perturbations of restricted shifts, J. Analyse Math, Volume 25 (1972), pp. 169-191 | DOI | MR | Zbl

[3] E. Fricain Bases of reproducing kernels in model spaces, J. Operator Theory, Volume 46 (2001), pp. 517-543 | MR | Zbl

[4] P. Gorkin; H.-M. Lingenberg; R. Mortini Homeomorphic disks in the spectrum of H , Indiana Univ. Math. J, Volume 39 (1990), pp. 961-983 | DOI | MR | Zbl

[5] H. Hedenmalm Thin interpolating sequences and three algebras of bounded functions, Proc. Amer. Math. Soc, Volume 99 (1987), pp. 489-495 | DOI | MR | Zbl

[6] E. Hewitt; K. Stromberg Real and Abstract Analysis, Springer-Verlag, Berlin-Heidelberg-New York, 1969 | MR | Zbl

[7] S.V. Hru#x0161;#x010D;ev; N.K. Nikolski; B.S. Pavlov; V.P. Havin and N.K. Nikolski, Eds. Unconditional bases of exponentials and of reproducing kernels, Complex Analysis and Spectral Theory (Lecture Notes in Mathematics) (1981), pp. 214-335 | Zbl

[8] N. Levinson Gap and Density Theorems (Colloquium Publ.), Volume 26 (1940) | JFM | Zbl

[9] N.K. Nikolski Treatise on the Shift Operator, Grundlehren der mathematischen Wissenschafte, vol. 273, Springer-Verlag, Berlin, 1986 | MR | Zbl

[10] N.K. Nikolski Operators, Functions and Systems: An Easy Reading. Model Operators and Systems, Mathematical Surveys and Monographs, 93, Volume 2, American Mathematical Society, 2002 | MR | Zbl

[11] N.K. Nikolski; A.L. Volberg; C. Sadosky Ed. Tangential and approximate free interpolation, Analysis and partial differential equations (Lecture Notes in Pure and Applied. Math.) (1990), pp. 277-299 | Zbl

[12] R.E. Paley; N. Wiener Fourier Transforms in the Complex Domain, Colloquium Publ., vol. 19, Amer. Math. Soc., Providence, 1934 | Zbl

[13] C. Sundberg; T.H. Wolff Interpolating sequences for QA B , Trans. Amer. Math. Soc, Volume 276 (1983), pp. 551-581 | MR | Zbl

[14] B. Sz-Nagy; C. Foias Harmonic Analysis of Operators on Hilbert Spaces, North-Holland Publishing Co., Amsterdam-London, 1970 | MR | Zbl

[15] A.L. Volberg Two remarks concerning the theorem of S. Axler, S.-Y. A. Chang and D. Sarason, J. Operator Theory, Volume 7 (1982), pp. 209-218 | MR | Zbl

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