Completeness of reproducing kernels in the model spaces

Emmanuel Fricain[1]

  • [1] Université Claude Bernard Lyon I, Institut Girard Desargues, 43 bd du 11 novembre 1918, 69622 Villeurbanne Cedex (France)

Annales de l’institut Fourier (2002)

  • Volume: 52, Issue: 2, page 661-686
  • ISSN: 0373-0956

Abstract

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Let ( λ n ) n ≥ 1 be a Blaschke sequence of the unit disc 𝔻 and Θ be an inner function. Assume that the sequence of reproducing kernels k Θ ( z , λ n ) : = 1 - Θ ( λ n ) ¯ Θ ( z ) 1 - λ n ¯ z n ≥ 1 is complete in the model space K Θ p : = H p ∩ Θ H 0 p ¯ , 1 < p < + ∞ . First of all, we study the stability of this completeness not only under perturbations of frequencies ( λ n ) n ≥ 1 but also under perturbations of function Θ . We recover some classical results on exponential systems. Then, if we assume further that the sequence ( k Θ ( . , λ n ) ) n ≥ 1 is minimal, we show that, for a certain class of functions Θ , the biorthogonal family is also complete.

How to cite

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Fricain, Emmanuel. "Complétude des noyaux reproduisants dans les espaces modèles." Annales de l’institut Fourier 52.2 (2002): 661-686. <http://eudml.org/doc/115990>.

@article{Fricain2002,
abstract = {Soit $(\lambda _n)_\{n\ge 1\}$ une suite de Blaschke du disque unité $\{\mathbb \{D\}\}$ et $\Theta $ une fonction intérieure. On suppose que la suite de noyaux reproduisants $\Big (k_\Theta (z,\lambda _n):= \{1-\overline\{\Theta (\lambda _n)\}\Theta (z)\over 1- \overline\{\lambda _n\}z\}\Big )_\{n\ge 1\}$ est complète dans l’espace modèle $K_\Theta ^p:=H^p\cap \Theta \overline\{H^p_0\}$, $1&lt;p&lt;+\infty $. On étudie, dans un premier temps, la stabilité de cette propriété de complétude, à la fois sous l’effet de perturbations des fréquences $(\lambda _n)_\{n\ge 1\}$ mais également sous l’effet de perturbations de la fonction $\Theta $. On retrouve ainsi un certain nombre de résultats classiques sur les systèmes d’exponentielles. Puis, si on suppose de plus que la suite $(k_\Theta (.,\lambda _n))_\{n\ge 1\}$ est minimale, on montre que, pour une certaine classe de fonctions $\Theta $, la famille biorthogonale associée est aussi complète.},
affiliation = {Université Claude Bernard Lyon I, Institut Girard Desargues, 43 bd du 11 novembre 1918, 69622 Villeurbanne Cedex (France)},
author = {Fricain, Emmanuel},
journal = {Annales de l’institut Fourier},
keywords = {Hardy spaces; reproducing kernels; completeness; exponential systems},
language = {fre},
number = {2},
pages = {661-686},
publisher = {Association des Annales de l'Institut Fourier},
title = {Complétude des noyaux reproduisants dans les espaces modèles},
url = {http://eudml.org/doc/115990},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Fricain, Emmanuel
TI - Complétude des noyaux reproduisants dans les espaces modèles
JO - Annales de l’institut Fourier
PY - 2002
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 2
SP - 661
EP - 686
AB - Soit $(\lambda _n)_{n\ge 1}$ une suite de Blaschke du disque unité ${\mathbb {D}}$ et $\Theta $ une fonction intérieure. On suppose que la suite de noyaux reproduisants $\Big (k_\Theta (z,\lambda _n):= {1-\overline{\Theta (\lambda _n)}\Theta (z)\over 1- \overline{\lambda _n}z}\Big )_{n\ge 1}$ est complète dans l’espace modèle $K_\Theta ^p:=H^p\cap \Theta \overline{H^p_0}$, $1&lt;p&lt;+\infty $. On étudie, dans un premier temps, la stabilité de cette propriété de complétude, à la fois sous l’effet de perturbations des fréquences $(\lambda _n)_{n\ge 1}$ mais également sous l’effet de perturbations de la fonction $\Theta $. On retrouve ainsi un certain nombre de résultats classiques sur les systèmes d’exponentielles. Puis, si on suppose de plus que la suite $(k_\Theta (.,\lambda _n))_{n\ge 1}$ est minimale, on montre que, pour une certaine classe de fonctions $\Theta $, la famille biorthogonale associée est aussi complète.
LA - fre
KW - Hardy spaces; reproducing kernels; completeness; exponential systems
UR - http://eudml.org/doc/115990
ER -

References

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  1. P. Ahern, D. Clark, Radial limits and invariant subspaces, Amer. of Math. 2 (1970), 332-342 Zbl0197.39202MR262511
  2. A. Beurling, P. Malliavin, On Fourier transforms of measures with compact support, Acta Math. 107 (1962), 291-309 Zbl0127.32601MR147848
  3. A. Beurling, P. Malliavin, On the closure of characters and the zeros of entire functions, Acta Math. 118 (1967), 79-93 Zbl0171.11901MR209758
  4. L.A. Coburn, Weyl's theorem for nonnormal operators, Michigan Math J. 13 (1966), 285-288 Zbl0173.42904MR201969
  5. K.C. Chan, S.M. Seubert, Reducing subspaces of compressed analytic Toeplitz operators on the Hardy space, Integr. Equa. Oper. Theory 28 (1997), 147-157 Zbl0907.47016MR1451498
  6. N. Danikas, On an identity theorem in the Nevanlinna class N , J. Approx. Theory 77 (1994), 184-190 Zbl0804.30031MR1275934
  7. R.G. Douglas, H.S. Shapiro, A.L. Shields, Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst. Fourier (Grenoble) (1970), 37-76 Zbl0186.45302MR270196
  8. P. Duren, Theory of H p Spaces, (1970), Academic Press, New-York Zbl0215.20203MR268655
  9. K.M. Dyakonov, Kernels of Toeplitz operators via Bourgain's factorization theorem Zbl0947.47019
  10. K.M. Dyakonov, Interpolating functions of minimal norm, star-invariant subspaces, and kernels of Toeplitz operators, Proc. Amer. Math. Soc. 116 (1992), 1007-1013 Zbl0769.30022MR1100649
  11. K.M. Dyakonov, Entire functions of exponentials type and model subspaces in H p , Journal of Math. Sciences 71 (1994), 2222-2233 MR1111913
  12. K.M. Dyakonov, Smooth functions in the range of a Hankel operator, Indiana Univ. Math. J. 43 (1994), 805-838 Zbl0821.30026MR1305948
  13. E. Fricain, Propriétés géométriques des suites de noyaux reproduisants dans les espaces modèles, (1999) 
  14. J.B. Garnett, Bounded analytic functions, (1981), Academic Press, New York Zbl0469.30024MR628971
  15. V.I. Gurarii, M.A. Meletidi, Stability of completeness of sequences in Banach spaces (Russian), Bull. Acad. Pol. Sci. 18 (1970), 533-536 Zbl0204.43304MR278039
  16. E. Hayashi, The solution sets of extremal problems in H 1 , Proc. Amer. Math. Soc. 93 (1985), 690-696 Zbl0565.30031MR776204
  17. E. Hayashi, The kernel of a Toeplitz operator, Integral Equations Operator Theory 9 (1986), 588-591 Zbl0636.47023MR853630
  18. E. Hayashi, Classification of nearly invariant subspaces of the backward shift, Proc. Amer. Math. Soc. 110 (1990), 441-448 Zbl0716.30028MR1019277
  19. W.K. Hayman, Identity theorems for functions of bounded characteristic, J. London Math. Soc. 58 (1998), 127-140 Zbl0922.30005MR1666094
  20. S. Hruschev, N. Nikolski, B. Pavlov, Unconditional bases of exponentials and reproducing kernels, 864 (1981), 214-335, Springer-Verlag, Berlin-Heidelberg Zbl0466.46018
  21. V.P. Havin, S.A. Vinogradov, Free interpolation in H ∞ and in some other function classes, Seminarov Mat. Inst. Steklova (LOMI) 47 (1974), 15-54 Zbl0355.41005
  22. S.Ya. Khavinson, Extremal problems for bounded analytic functions with interior side conditions, Russ. Math. Survey 18 (1963), 21-96 Zbl0149.03302
  23. P. Koosis, Leçons sur le Théorème de Beurling et Malliavin, Les Publications CRM, Montréal (1996) Zbl0869.30023MR1430571
  24. I.F. Krasichkov-Ternovskii, An interpretation of the Beurling-Malliavin theorem on the radius of completeness, Math. USSR Sbornik 66 (1990), 405-429 Zbl0698.30003MR993233
  25. N. Levinson, Gap and density Theorems, 26 (1940), Colloquium Publ., New-York Zbl0145.08003MR3208
  26. M. Lee, D.E. Sarason, The spectra of some Toeplitz operators, J. Math. Anal. Appl. 33 (1971), 529-543 Zbl0185.38401MR275203
  27. Y.I. Lyubarskii, K. Seip, A uniqueness theorem for bounded analytic functions, Bull. London Math. Soc. 29 (1997), 49-52 Zbl0889.30005MR1416406
  28. N.K. Nikolskii, Treatise on the shift operator, 273 (1986), Springer Verlag, Berlin-Heidelberg-New-York Zbl0587.47036MR827223
  29. R.M. Redheffer, Elementary remarks on completeness, Duke Math. J. 35 (1968), 103-116 Zbl0195.42502MR225090
  30. R.M. Redheffer, Completeness of sets of complex exponentials, Adv. in Math. 24 (1977), 1-62 Zbl0358.42007MR447542
  31. D. Sarason, Kernels of Toeplitz operators, Oper. Theory, Adv. Appl. 71 (1994), 153-164 Zbl0818.47025MR1300218
  32. R.M. Young, On complete biorthogonal systems, Proceedings of the Amer. Soc. 83 (1981), 537-540 Zbl0484.42009MR627686
  33. T. Kato, Perturbation theory for linear operators, 144 (1967), Springer Verlag, Berlin-Heidelberg-New-York Zbl0342.47009
  34. V.P. Havin, S.A. Vinogradov, Free interpolation in H ∞ and in some other function classes, Seminarov Mat. Inst. Steklova (LOMI) 9 (1978), 137-171 Zbl0398.41002

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