Skip to main content
SpringerLink
Log in

On a class of selfadjoint operators in Krein space and their compact perturbations

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

For a class of selfadjoint operators in a Krein space containing the definitizable selfadjoint operators a funetional calculus and the spectral function are studied. Stability properties of the spectral function with respect to small compact perturbations of the resolvent are proved.

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. Azizov, T.Ja.; Iohvidov, I.S.: Foundations of the theory of linear operators in spaces with an indefinite metric, Nauka, Moscou 1986 (Russian).

    Google Scholar 

  2. Bognár, J.: Indefinite inner product spaces, Springer-Verlag, Berlin-Heidelberg-New York 1974.

    Google Scholar 

  3. Colojoarâ, I.; Foias, C.: Theory of generalized spectral operators, Gordon and Breach, New York-London-Paris 1968.

    Google Scholar 

  4. Gohberg, I.C.; Krein, M.G.: Introduction to the theory of linear nonselfadjoint operators, Nauka, Moscou 1965 (Russian).

    Google Scholar 

  5. Iohvidov, I.S.; Krein, M.G.: Spectral theory of operators in spaces with an indefinite metric, I: Trudy Moskov. Mat. Obšč. 5 (1956), 367–432, II: Trudy Moskov. Mat. Obšč. 8 (1959), 413–496 (Russian).

    Google Scholar 

  6. Jonas, P.: Zur Existenz von Eigenspektralfunktionen mit Singularitäten, Math. Nachr. 88 (1977), 345–361.

    Google Scholar 

  7. Jonas, P.: On the functional calculus and the spectral function for definitizable operators in Krein space, Beiträge Anal. 16 (1981), 121–135.

    Google Scholar 

  8. Jonas, P.: On a class of unitary operators in Krein space, Operator Theory: Advances and Applications, Vol. 17, Birkhäuser Verlag, Basel-Boston-Stuttgart 1986, 151–172.

    Google Scholar 

  9. Jonas, P.: Die Spurformel der Störungstheorie für einige Klassen unitärer und selbstadjungierter Operatoren im Kreinraum, Akad. Wiss. DDR, Karl-Weierstrass-Institut für Mathematik, Report R-Math-06/86, 1986.

  10. Kitano, K.: The growth of the resolvent and hyperinvariant subspaces, Tôhoku Math. J. 25 (1973), 317–331.

    Google Scholar 

  11. Komatsu, H.: Ultradistributions, I, Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo, Sect. I A, Math. 20 (1973), 25–105.

    Google Scholar 

  12. Köthe, G.: Topologische lineare Räume I, Springer-Verlag, Berlin-Göttingen-Heidelberg 1960.

    Google Scholar 

  13. Krein, M.G.: A new application of the fixed-point principle in the theory of operators in a space with indefinite metric, Dokl. Akad. Nauk SSSR 154 (1964), 1023–1026 (Russian).

    Google Scholar 

  14. Krein, M.G.: Introduction to the geometry of indefinite J-spaces and to the theory of operators in these spaces. In: Second mathematical summer school, Part I, pp. 15–92, Naukova Dumka, Kiev 1965 (Russian).

    Google Scholar 

  15. Krein, M.G.; Langer, H.: Über dieQ-Funktion eines π-hermiteschen Operators im Raume πϰ, Acta Scient. Math. (Szeged) 34, (1973), 191–230.

    Google Scholar 

  16. Langer, H.: Eine Verallgemeinerung eines Satzes von L.S. Pontrjagin, Math. Annalen 152 (1963), 434–436.

    Google Scholar 

  17. Langer, H. Spektraltheorie linearer Operatoren in J-Räumen und einige Anwendungen auf die Schar L(λ) = λ2I + λB + C, Habilitations-schrift, Dresden 1965.

  18. Langer, H.: Spektralfunktionen einer Klasse J-selbstadjungierter Operatoren, Math. Nachr. 33 (1967), 107–120.

    Google Scholar 

  19. Langer, H.: Spectral functions of definitizable operators in Krein spaces, Functional analysis, Proceedings of a conference held at Dubrovnik, Lecture Notes in Mathematics 948, Springer-Verlag, Berlin-Heidelberg-New York 1982, 1–46.

    Google Scholar 

  20. Langer, H.; Najman, B.: Perturbation theory for definitizable operators in Krein spaces, J. Operator Theory 9 (1983), 297–317.

    Google Scholar 

  21. Ljubič, Ju.I.; Macaev, V.I.: On operators with separable spectrum, Mat. Sb. 56 (1962), 433–468 (Russian).

    Google Scholar 

  22. Nagy, B.: Operators with spectral singularities, J. Operator Theory 15 (1986), 307–325.

    Google Scholar 

  23. Reed, M.; Simon, B.: Methods of modern mathematical physics. I, Academic Press, New York-London 1972.

    Google Scholar 

  24. Roumieu, Ch.: Ultra-distributions définies sur Rn et sur certaines classes de variétés différentiables, J. Anal. Math. 10 (1962/63), 153–192.

    Google Scholar 

  25. Zsidó, L.: Invariant subspaces of compact perturbations of linear operators in Banach spaces, J. Operator Theory 1 (1979), 225–260.

    Google Scholar 

  26. Bart, H.; Gohberg, I.; Kaashoek, M.A.: Minimal factorization of matrix and operator functions, Operator theory: Advances and applications, Vol.1, Birkhäuser Verlag, Basel-Boston-Stuttgart 1979.

    Google Scholar 

  27. Gohberg, I.C.; Sigal, E.I.: An operator generalization of the logarithmic residue theorem and the theorem of Rouché, Mat. Sbornik 84 (1971), 607–629 (Russian).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Akademie der Wissenschaften, DDR-1086, Berlin, GDR

    Peter Jonas

Authors
  1. Peter Jonas
    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

Jonas, P. On a class of selfadjoint operators in Krein space and their compact perturbations. Integr equ oper theory 11, 351–384 (1988). https://doi.org/10.1007/BF01202078

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01202078

Keywords

Search

Navigation