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Recent Advances in Operator Theory in Hilbert and Krein Spaces pp 125–169Cite as

Approximation of N ∞ κ -functions II: Convergence of Models

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Part of the book series: Operator Theory: Advances and Applications ((OT,volume 198))

Abstract

This paper is a continuation of Part I, au][9]_in the list of references, where models for N ∞к -functions have been studied in detail. In the present paper we investigate the convergence of the corresponding models as a singular N ∞к -function is approximated by regular N ∞к -functions. This involves the theory about approximating an operator by operators acting in different spaces. In the last section an example related to the Bessel differential operator is worked out.

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References

  1. L. de Branges, Hilbert spaces of entire functions, Prentice-Hall, Englewood Cliffs, N.J., 1968.

    MATH  Google Scholar 

  2. R.B. Burckel, An introduction to classical complex analysis, Vol. 1, Mathematische Reihe 64, Birkhäuser, Basel, 1979.

    Google Scholar 

  3. E.A. Coddington and R.G. Gilbert, Generalized resolvents of ordinary differential operators, Trans. Amer. Math. Soc. 93(2) (1959), 216–241.

    Article  MATH  MathSciNet  Google Scholar 

  4. E.A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955.

    MATH  Google Scholar 

  5. A. Dijksma, H. Langer, A. Luger, and Yu. Shondin, Minimal realizations of scalar generalized Nevanlinna functions related to their basic factorization, Oper. Theory Adv. Appl. 154, Birkhäuser, Basel, 2004, 69–90.

    Google Scholar 

  6. A. Dijksma, H. Langer, and H.S.V. de Snoo, SelfadjointΠκ-extensions of symmetric subspaces: an abstract approach to boundary problems with spectral parameter in the boundary conditions, Integral Equations Operator Theory 7(4) (1984), 459–515; Addendum: Integral Equations Operator Theory 7(6) (1984), 905.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Dijksma, H. Langer, Yu. Shondin, and C. Zeinstra, Self-adjoint operators with inner singularities and Pontryagin spaces, Oper. Theory Adv. Appl. 118, Birkhäuser, Basel, 2000, 105–175.

    Google Scholar 

  8. A. Dijksma, A. Luger, and Yu. Shondin, Minimal models for N ∞ κ -functions, Oper. Theory Adv. Appl. 163, Birkhäuser, Basel, 2005, 97–134.

    Google Scholar 

  9. A. Dijksma, A. Luger, and Yu. Shondin, Approximation of N ∞ κ -functions I: models and regularization, Oper. Theory Adv. Appl. 188, Birkhäuser, Basel, 2008, 87–112.

    Google Scholar 

  10. A. Dijksma and Yu. Shondin, Singular point-like perturbations of the Bessel operator in a Pontryagin space, J. Differential Equations 164 (2000), 49–91.

    Article  MATH  MathSciNet  Google Scholar 

  11. C. Fulton, Titchmarsh-Weyl m-functions for Second-order Sturm-Liouville Problems with two singular endpoints, Math. Nachr. 281 (2008), 1418–1475.

    Article  MATH  MathSciNet  Google Scholar 

  12. F. Gesztesy and M. Zinchenko, On spectral theory for Schrödinger operators with strongly singular potentials, Math. Nachr. 279(9–10) (2006), 1041–1082.

    Article  MATH  MathSciNet  Google Scholar 

  13. T. Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften 132, Springer, Heidelberg, 1966.

    Google Scholar 

  14. P. Kurasov and A. Luger, An operator theoretic interpretation of the generalized Titchmarsh-Weyl coefficient for a singular Sturm-Liouville problem, submitted.

    Google Scholar 

  15. H. Langer and A. Luger, Convergence of generalized Nevanlinna functions, in preparation.

    Google Scholar 

  16. B.M. Levitan and I.S. Sargsjan, Introduction to Spectral theory, Transl. Math. Monographs 39, Amer. Math. Soc., Providence, R.I., 1975.

    Google Scholar 

  17. H. Langer and B. Najman, Perturbation theory for definizable operators in Krein spaces, J. Operator Theory 9 (1983), 247–317.

    MathSciNet  Google Scholar 

  18. B. Najman, Perturbation theory for selfadjoint operators in Pontrjagin spaces, Glasnik Mat. 15 (1980) 351–370.

    MathSciNet  Google Scholar 

  19. Yu. Shondin, On approximation of high order singular perturbations, J. Phys. A: Math. Gen. 38 (2005), 5023–5039.

    Article  MATH  MathSciNet  Google Scholar 

  20. O.Yu. Shvedov, Approximations for strongly singular evolution equations, J. Funct. Anal. 210(2) (2004), 259–294.

    Article  MATH  MathSciNet  Google Scholar 

  21. B. Simon, A canonical decomposition for quadratic forms with applications to monotone convergence theorems, J. Funct. Anal. 28(3) (1978), 377–385.

    Article  MATH  MathSciNet  Google Scholar 

  22. F. Stummel, Diskrete Konvergenz linearer Operatoren I, Math. Annal. 190 (1970), 45–92; II, Math. Z. 141 (1975), 231–264.

    Google Scholar 

  23. V.A. Yavrian, On M.G. Krein’s spectral shift function for canonical systems of differential equations, Oper. Theory Adv. Appl. 117, Birkhäuser, Basel, 2000, 393–417.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Groningen, P.O. Box 407, 9700 AK, Groningen, The Netherlands

    Aad Dijksma

  2. Department of Mathematics, Lund Institute of Technology, Box 118, SE-221 00, Lund, Sweden

    Annemarie Luger

  3. Department of theoretical Physics, State Pedagogical University, Str. Ulyanova 1, Nizhny Novgorod, GSP 37, 603950, Russia

    Yuri Shondin

Authors
  1. Aad Dijksma
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  2. Annemarie Luger
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  3. Yuri Shondin
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Editor information

Editors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Jussi Behrndt  & Karl-Heinz Förster  & 

  2. Institut für Mathematik, Technische Universität Ilmenau, Postfach 100565, 98684, Ilmenau, Germany

    Carsten Trunk

Additional information

Dedicated to Peter Jonas, in memoriam

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Dijksma, A., Luger, A., Shondin, Y. (2009). Approximation of N ∞ κ -functions II: Convergence of Models. In: Behrndt, J., Förster, KH., Trunk, C. (eds) Recent Advances in Operator Theory in Hilbert and Krein Spaces. Operator Theory: Advances and Applications, vol 198. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0180-1_8

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