Skip to main content
SpringerLink
Log in

On Schur–Nevanlinna–Pick Indefinite Interpolation Problem

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

The Schur–Nevanlinna–Pick interpolation problem is considered in the class of generalized Schur functions and reduced to the problem of the extension of a certain isometric operator V that acts in the Pontryagin space. The description of the solutions of this problem is based on the theory of the resolvent matrix developed by Krein.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. M. G. Krein and H. Langer, “Ñber die verallgemeinerten resolventen und die characteristische function eines isometrischen operators im raume Пκ,” in: Hilbert Space Operators and Operator Algebras, Colloquium of the Janos Bolyai Mathematical Society, Vol. 5, North Holland, Amsterdam (1972), pp. 353–399.

    Google Scholar 

  2. V. M. Adamyan, D. Z. Arov, and M. G. Krein, “Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Tagaki problem,” Mat. Sb., 86, No. 1, 43-75 (1971).

    Google Scholar 

  3. I. P. Fedchina, “A criterion for the solvability of a tangential Nevanlinna-Pick problem,” Mat. Issled., 26, 213-226 (1972).

    Google Scholar 

  4. A. A. Nudel'man, “On one generalization of classical interpolation problems,” Dokl. Akad. Nauk SSSR, 256, 790-793 (1981).

    Google Scholar 

  5. J. A. Ball, L. Gohberg, and L. Rodman, “Interpolation of rational matrix functions,” Oper. Theory: Adv. Appl., 45, 1-72 (1988).

    Google Scholar 

  6. D. Alpay, P. Bruinsma, A. Dijksma, and H. S. V. de Snoo, “Interpolation problems, extensions of symmetric operators and reproducing kernel Pontryagin spaces. I,” Oper. Theory: Adv. Appl., 50, 35-82 (1991).

    Google Scholar 

  7. A. Amirshadyan and V. Derkach, “Interpolation in generalized Nevanlinna and Stieltjes classes,” J. Operator Theory, 42, 145-188 (1999).

    Google Scholar 

  8. D. Sarason, “Generalized interpolation in H ∞,” Trans. Amer. Math. Soc., 127, 179-203 (1967).

    Google Scholar 

  9. J. A. Ball and J. W. Helton, “A Beurling-Lax theorem for the Lie group U(m, n) which contains most classical interpolation theory,” J. Operator Theory, 9, 107-142 (1983).

    Google Scholar 

  10. J. A. Ball and J. W. Helton, “Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions,” Integr. Equat. Oper. Theory, 9, 155-203 (1986).

    Google Scholar 

  11. A. Gheondea, “On generalized interpolation and shift invariant maximal semidefinite subspaces,” Oper. Theory: Adv. Appl., 103, 121-136 (1998).

    Google Scholar 

  12. D. Z. Arov, “Generalized bitangent Caratheodory-Nevanlinna-Pick problem and (j,J 0)-inner matrix functions,” Izv. Ross. Akad. Nauk, Ser. Mat., 57, No. 1 (1993).

    Google Scholar 

  13. A. Ya. Kheifets, “Generalized bitangent Schur-Nevanlinna-Pick problem and the Parseval equality related to it,” Teor. Funkts. Funkts. Anal. Prilozhen. (Kharkov), 54, 89-96 (1990).

    Google Scholar 

  14. V. É. Katsnel'son, A. Ya. Kheifets, and P. M. Yuditskii, “Abstract interpolation problem and the theory of extensions of isometric operators,” in: Operators in Functional Spaces and Problems in the Theory of Functions [in Russian], Naukova Dumka, Kiev (1987), pp. 83-96.

    Google Scholar 

  15. V. A. Derkach, “On indefinite abstract interpolation problem,” Meth. Funct. Anal. Topol., 7, No. 4, 87-100 (2001).

    Google Scholar 

  16. M. G. Krein, “On resolvents of Hermitian operators with deficiency index (m, m),” Dokl. Akad. Nauk SSSR, 52, No. 8, 657-660 (1946).

    Google Scholar 

  17. M. G. Krein and Sh. N. Saakyan, “On new results in the theory of resolvents of Hermitian operators,” Dokl. Akad. Nauk SSSR, 169, No. 6, 1269-1272 (1966).

    Google Scholar 

  18. D. Alpay, A. Dijksma, J. Rovnyak, and H. S. V. de Snoo, “Schur functions, operator colligations, and reproducing kernel Pontryagin spaces,” Oper. Theory: Adv. Appl., 96 (1997).

  19. A. Dijksma, H. Langer, and H. S. V. de Snoo, “Characteristic functions of unitary colligations in Пκ-spaces,” Oper. Theory: Adv. Apply., 19, 125–194 (1986).

    Google Scholar 

  20. B. Szökefalvi-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space [Russian translation], Mir, Moscow (1970).

    Google Scholar 

  21. M. G. Krein, “Fundamental aspects of the theory of representations of Hermitian operators with deficiency indices (n,n),” Ukr. Mat. Zh., 1, No. 2, 3-66 (1949).

    Google Scholar 

  22. V. A. Derkach, “On generalized resolvents of Hermitian relations in Krein spaces,” J. Math. Sci., 97, No. 5, 4420-4460 (1999).

    Google Scholar 

  23. V. A. Derkach and M. M. Malamud, “The extension theory of Hermitian operators and the moment problem,” J. Math. Sci., 73, 141-242 (1995).

    Google Scholar 

  24. V. Derkach, On Indefinite Generalized Interpolation, Preprint No. 262, University of Helsinki, Helsinki (2000).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Donetsk National University, Donetsk

    V. A. Derkach

Authors
  1. V. A. Derkach
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Derkach, V.A. On Schur–Nevanlinna–Pick Indefinite Interpolation Problem. Ukrainian Mathematical Journal 55, 1567–1587 (2003). https://doi.org/10.1023/B:UKMA.0000022069.69507.bc

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:UKMA.0000022069.69507.bc

Keywords

Search

Navigation