Geometric and Functional Analysis

, Volume 25, Issue 2, pp 417–452 | Cite as

Spectral synthesis in de Branges spaces



We solve completely the spectral synthesis problem for reproducing kernels in the de Branges spaces \({\mathcal{H}(E)}\) . Namely, we describe the de Branges spaces \({\mathcal{H}(E)}\) such that every complete and minimal system of reproducing kernels \({\{k_\lambda\}_{\lambda \in \Lambda}}\) with complete biorthogonal \({\{g_\lambda\}_{\lambda \in \Lambda}}\) admits the spectral synthesis, i.e., \({f \in \overline{\rm Span} \{(f, g_\lambda) k_\lambda : \lambda \in \Lambda\}}\) for any f in \({\mathcal{H}(E)}\) . Surprisingly, this property takes place only for two essentially different classes of de Branges spaces: spaces with finite spectral measure and spaces which are isomorphic to Fock-type spaces of entire functions. The first class goes back to de Branges himself, while special cases of de Branges spaces of the second class appeared in the literature only recently; we give a complete characterisation of this second class in terms of the spectral data for \({\mathcal{H}(E)}\) .


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. AS93.
    Azoff E., Shehada H.: Algebras generated by mutually orthogonal idempotent operators. The Journal of Operator Theory, 29, 249–267 (1993)MATHMathSciNetGoogle Scholar
  2. BB11.
    Baranov A., Belov Yu.: Systems of reproducing kernels and their biorthogonal: completeness or incompleteness?. International Mathematics Research Notices, 22, 5076–5108 (2011)MathSciNetGoogle Scholar
  3. BBB13.
    Baranov A., Belov Yu., Borichev A.: Hereditary completeness for systems of exponentials and reproducing kernels. Advances in Mathematics, 235, 525–554 (2013)CrossRefMATHMathSciNetGoogle Scholar
  4. BBBY13.
    A. Baranov, Yu. Belov, A. Borichev and D. Yakubovich. Recent developments in spectral synthesis for exponential systems and for non-self-adjoint operators. In: Recent Trends in Analysis (Proceedings of the Conference in Honor of Nikolai Nikolski, Bordeaux, 2011). Theta Foundation, Bucharest (2013), pp. 17–34Google Scholar
  5. BY15.
    A.D. Baranov and D.V. Yakubovich, Completeness and spectral synthesis of nonselfadjoint one-dimensional perturbations of selfadjoint operators. arXiv:1212.5965.
  6. BMS10.
    Belov Yu., Mengestie T., Seip K.: Unitary discrete Hilbert transforms. Journal d’Analyse Mathématique, 112, 383–395 (2010)CrossRefMATHMathSciNetGoogle Scholar
  7. BMS11.
    Belov Yu., Mengestie T., Seip K.: Discrete Hilbert transforms on sparse sequences. Proceedings of the London Mathematical Society, 103, 73–105 (2011)CrossRefMATHMathSciNetGoogle Scholar
  8. BL10.
    Borichev A., Lyubarskii Yu.: Riesz bases of reproducing kernels in Fock type spaces. Journal of the Institute of Mathematics of Jussieu, 9, 449–461 (2010)CrossRefMATHMathSciNetGoogle Scholar
  9. Bra68.
    L. de Branges, Hilbert Spaces of Entire Functions. Prentice–Hall, Englewood Cliffs (1968)Google Scholar
  10. Cla72.
    Clark D.N.: One-dimensional perturbations of restricted shifts. Journal d’Analyse Mathématique, 25, 169–191 (1972)CrossRefMATHGoogle Scholar
  11. DN76.
    L. Dovbysh and N. Nikolski. Two methods for avoiding hereditary completeness. Zapiski Nauchnykh Seminarov LOMI, 65 (1976), 183–188; English transl.: Journal of Soviet Mathematics 16 (1981), 1175–1179Google Scholar
  12. DNS86.
    L. Dovbysh, N. Nikolski and V. Sudakov. How good can a nonhereditary family be? Zapiski Nauchnykh Seminarov LOMI 73 (1977), 52–69; English transl.: Journal of Soviet Mathematics, 34 (1986), 2050–2060Google Scholar
  13. Ham51.
    Hamburger H.: Über die Zerlegung des Hilbertschen Raumes durch vollstetige lineare Transformationen. Mathematische Nachrichten 4, 56–69 (1951)MATHMathSciNetGoogle Scholar
  14. HJ94.
    V. Havin and B. Jöricke. The Uncertainty Principle in Harmonic Analysis. Springer, Berlin (1994)Google Scholar
  15. HV96.
    Holt J., Vaaler J.: The Beurling–Selberg extremal functions for a ball in Euclidean space. Duke Mathematical Journal, 83, 202–248 (1996)CrossRefMathSciNetGoogle Scholar
  16. KLP93.
    Katavolos A., Lambrou M., Papadakis M.: On some algebras diagonalized by M-bases of \({\ell^2}\) . Integral Equations and Operator Theory 17, 68–94 (1993)CrossRefMATHMathSciNetGoogle Scholar
  17. Lag06.
    J. Lagarias. Hilbert spaces of entire functions and Dirichlet L-functions. In: Frontiers in Number Theory, Physics, and Geometry. I. Springer, Berlin (2006), pp. 365–377Google Scholar
  18. LW90.
    Larson D., Wogen W.: Reflexivity properties of \({T \oplus 0}\). Journal of Functional Analysis, 92, 448–467 (1990)CrossRefMATHMathSciNetGoogle Scholar
  19. MP05.
    N. Makarov and A. Poltoratski. Meromorphic inner functions, Toeplitz kernels and the uncertainty principle. Perspectives in Analysis Math. Phys. Stud., Vol. 27. Springer, Berlin (2005), pp. 185–252Google Scholar
  20. Mar70.
    A. Markus. The problem of spectral synthesis for operators with point spectrum. Izvestiya Akademii Nauk SSSR, 34 (1970), 662–688; English transl.: Mathematics of the USSR-Izvestiya, 4 (1970), 670–696Google Scholar
  21. MP10.
    Mitkovski M., Poltoratski A.: Pólya sequences, Toeplitz kernels and gap theorems. Advances in Mathematics, 224, 1057–1070 (2010)CrossRefMATHMathSciNetGoogle Scholar
  22. Nik69.
    N. Nikolski. Complete extensions of Volterra operators. Izv. Akad. Nauk SSSR, 33 (1969), 1349–1353 (Russian); English transl.: Mathematics of the USSR-Izvestiya 3 (1969), 1271–1276Google Scholar
  23. Nik02.
    N. Nikolski. Operators, Functions, and Systems: an Easy Reading. Vol. 2, Math. Surveys Monogr., Vol. 93. AMS, Providence (2002)Google Scholar
  24. OS02.
    Ortega-Cerdà à J., Seip K.: Fourier frames. Annals of Mathematics 155, 789–806 (2002)CrossRefMathSciNetGoogle Scholar
  25. Wer52.
    Wermer J.: On invariant subspaces of normal operators. Proceedings of the American Mathematical Society 3, 270–277 (1952)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Anton Baranov
    • 1
    • 2
  • Yurii Belov
    • 3
  • Alexander Borichev
    • 4
  1. 1.Department of Mathematics and MechanicsSt. Petersburg State UniversitySt. PetersburgRussia
  2. 2.National Research University Higher School of EconomicsSt. PetersburgRussia
  3. 3.Chebyshev LaboratorySt. Petersburg State UniversitySt. PetersburgRussia
  4. 4.I2M, Aix-Marseille Université, CNRSMarseilleFrance

Personalised recommendations