Skip to main content
SpringerLink
Log in

On generalized resolvents of Hermitian operators in Krein spaces

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

Various classes of extensions and generalized resolvents of Hermitian operators acting in Krein spaces are described in terms of abstract boundary conditions.

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. T. Ya. Azizov and I. S. Iokhvidov, “Linear operators in spaces with indefinite metric and their applications,” in:Itogi VINITI, Ser. Mat. Analiz [in Russian], Vol. 17, VINITI, Moscow (1979), pp. 113–205.

    Google Scholar 

  2. V. I. Gorbachuk and M. L. Gorbachuk,Boundary-Value Problems for Operator Differential Equations, Kluwer, Dordrecht (1991).

    Google Scholar 

  3. V. A. Derkach, “Extensions of an Hermitian operator in a Krein space,”Dokl. Akad. Nauk Ukr.SSR., Ser. A, No. 5, 5–9 (1988).

    Google Scholar 

  4. M. G. Krein, “On the resolvents of an Hermitian operator with deficiency index (m, m),”Dokl. Akad. Nauk SSSR,52, No. 8, 657–660 (1946).

    Google Scholar 

  5. R. Arens, “Operational calculus of linear relations,”Pacif. J. Math.,11, No. 1, 9–23 (1961).

    Google Scholar 

  6. F. S. Rofe-Beketov, “On self-adjoint extensions of differential operators in a space of vector functions,”Teor. Funk., Funkts. Anal. Prilozhen., Issue 8, 3–24, (1969).

    Google Scholar 

  7. T. Kato,Perturbation Theory for Linear Operators, Springer, Berlin (1966).

    Google Scholar 

  8. M. G. Krein and H. Langer, “Über die Q-Funktion eines π-Hermiteschen Operators im Raume 1262-01,”Acta Sci. Math.,34, 191–230 (1973).

    Google Scholar 

  9. M. G. Krein and H. Langer, “Some propositions on analytic matrix functions related to the theory of operators in the space 1262-02,”Acta Sci. Math.,43, 181–205 (1981).

    Google Scholar 

  10. M. G. Krein and H. Langer, “On defect subspaces and generalized resolvents of an Hermitian operator in the space 1262-03,”Funkts. Anal.,5, No. 3, 54–69 (1971).

    Google Scholar 

  11. A. V. Shtraus, “On self-adjoint operators in the orthogonal sum of Hilbert spaces,”Dokl. Akad. Nauk SSSR,144, No. 3, 512–515 (1962).

    Google Scholar 

  12. V. M. Bruk, “On the invertible restrictions of closed operators in Banach spaces,” in:Functional Analysis. Spectral Theory [in Russian], Ulyanovsk (1988), pp. 32–42.

  13. A. V. Kuzhel', “Regular extensions of Hermitian operators in a space with indefinite metric,”Dokl. Akad. Nauk SSSR,265, No. 5, 1059–1061 (1982).

    Google Scholar 

  14. V. A. Derkach,Generalized Resolvents of Hermitian Operators in Krein Spaces [in Russian], Preprint No. 92.02, Institute of Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk (1992).

    Google Scholar 

  15. V. A. Derkach and M. M. Malamud,Weyl Functions of Hermitian Operators and Their Relationship to Characteristic Functions [in Russian], Preprint No. 85.09, Donetsk Physicotechnical Institute, Ukrainian Academy of Sciences, Donetsk (1985).

    Google Scholar 

  16. V. A. Derkach and M. M. Malamud, “On the Weyl function and Hermitian operators with gaps,”Dokl. Akad. Nauk SSSR,293, No. 5, 1041–1046 (1987).

    Google Scholar 

  17. V. A. Derkach and M. M. Malamud, “Generalized resolvents and the boundary value problems for Hermitian operators with gap,”J. Funct. Anal.,95, No. 1, 1–95 (1991).

    Google Scholar 

  18. I. Ts. Gokhberg and M. G. Krein,An Introduction to the Theory of Linear Nonself-Adjoint Operators in Hilbert Spaces [in Russian], Nauka, Moscow (1965).

    Google Scholar 

  19. T. Ya. Azizov, “To the theory of extensions of J-isometric and J-symmetric operators,”Funkts. Anal. Prilozhen.,18, Issue 1, 57–58 (1984).

  20. A. Dijksma, H. Langer, and H. de Snoo, “Unitary colligations in Krein spaces,”Lect. Notes Math.,1242, 1–42 (1987).

    Google Scholar 

  21. D. Z. Arov, “Passive linear stationary dynamic systems,”Sib. Mat. Zh.,20, No. 2, 211–228 (1979).

    Google Scholar 

  22. M. G. Krein and H. Langer, “Über einige Forsetzungsprobleme die eng mit der Theorie Hermitescher Operatoren im Raume 1262-04 zusammenhangen, I: Einige Funktionenklassen und ihre Darstellungen,”Math. Nachr.,77, 187–236 (1977).

    Google Scholar 

  23. A. V. Shtraus, “Extensions and generalized resolvents of a nondensely defined symmetric operator,”Izv. Akad. Nauk SSSR, Ser. Mat.,34, No. 1, 57–61 (1970).

    Google Scholar 

  24. A. A. Nikonov, “On generalized resolvents of densely defined operators in spaces with indefinite metric,” in:Functional Analysis [in Russian], Issue 2, Ulyanovsk (1974), pp. 42–45.

Download references

Author information

Authors and Affiliations

  1. Makeevka Institute of Civil Engineers, Makeevka

    V. A. Derkach

Authors
  1. V. A. Derkach
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1134–1147, September, 1994.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Derkach, V.A. On generalized resolvents of Hermitian operators in Krein spaces. Ukr Math J 46, 1248–1262 (1994). https://doi.org/10.1007/BF01059416

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01059416

Keywords

Search

Navigation