Skip to main content
SpringerLink
Log in

Representation theorems and variational principles for self-adjoint operator matrices

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

We use the notion of triples D+ → H → D− of Hilbert spaces to develop an analog of the Friedrichs extension procedure for a class of nonsemibounded operator matrices. In addition, we suggest a general approach (stated in the same terms) to the construction of variational principles for the eigenvalues of such matrices.

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. Kraus, M. Langer, and C. Tretter, “Variational principles and eigenvalue estimates for unbounded block operator matrices and applications,” J. Comput. Appl.Math. 171 (1-2), 311–334 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  2. J.-L. Lions and E. Magenes, Problèmes aux limites non-homogènes et applications, Vol. 1 (Dunod, Paris, 196;Mir, Moscow, 1971).

    MATH  Google Scholar 

  3. F. A. Berezin and M. A. Shubin, The Schrödinger Equation (Izd. Moskov. Univ., Moscow, 1983) [in Russian].

    MATH  Google Scholar 

  4. T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Heidelberg, 1966; Mir, Moscow, 1972).

    Book  MATH  Google Scholar 

  5. M. A. Naimark, Linear Differential Operators (Nauka, Moscow, 1969) [in Russian].

    MATH  Google Scholar 

  6. N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space (Nauka, Moscow, 1966) [in Russian].

    MATH  Google Scholar 

  7. A. A. Vladimirov, “Estimates of the number of eigenvalues of self-adjoint operator functions,” Mat. Zametki [Math. Notes] 74 (6), 838–847 (2003) [Math. Notes 74 (5–6), 794–802 (2003)].

    Article  MathSciNet  Google Scholar 

  8. V. A. Trenogin, Functional Analysis (Nauka, Moscow, 1980) [in Russian].

    MATH  Google Scholar 

  9. H. Langer, M. Langer, and C. Tretter, “Variational principles for eigenvalues of block operator matrices,” Indiana Univ. Math. J. 51 (6), 1427–1459 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Lancaster, A. Shkalikov, and Qiang Ye, “Strongly definitizable linear pencils in Hilbert space,” Integral Equations Operator Theory 17 (3), 338–360 (1993).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Moscow, Russia

    A. A. Vladimirov

  2. Dorodnicyn Computing Center of the Russian Academy of Sciences, Moscow, Russia

    A. A. Vladimirov

Authors
  1. A. A. Vladimirov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Vladimirov.

Additional information

Original Russian Text © A. A. Vladimirov, 2017, published in Matematicheskie Zametki, 2017, Vol. 101, No. 4, pp. 516–530.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vladimirov, A.A. Representation theorems and variational principles for self-adjoint operator matrices. Math Notes 101, 619–630 (2017). https://doi.org/10.1134/S0001434617030208

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434617030208

Keywords

Search

Navigation