Skip to main content
SpringerLink
Log in

Functional model for dissipative operators. A coordinate-free approach

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

Abstract

The paper has, basically, a methodical character. Dissipative operators are investigated with the aid of the Cayley contraction-transformation on the basis of the “coordinate-free model” of V. L Vasyunin and N. K. Nikol'skii. It is shown that the known forms of the characteristic function and of the selfadjoint dilation can be obtained as various realization of a general scheme. One obtains also new formulas for the dilation and its eigenfunctions, generalizing B. S. Pavlov's formulas for the Schrödinger operator on the semiaxis with a real potential and complex boundary condition. Examples are considered.

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

Literature cited

  1. N. K. Nikol'skii and V. I. Vasyunin, “A unified approach to function models, and the transcription problem,” Preprint LOMI E-5-86 (1986).

  2. B. S. Pavlov, “Dilation theory and spectral analysis of nonselfadjoint differential operators,” in: Mathematical Programming and Related Questions (Proc. Seventh Winter School, Drogobych, 1974) [in Russian], Tsentral. Ékonom. Mat. Inst. Akad. Nauk SSSR, Moscow (1976), pp. 3–69.

    Google Scholar 

  3. B. S. Pavlov, “Selfadjoint dilation of the dissipative Schrödinger operator and the expansion in its eigenfunctions,” Mat. Sb.,102 (144), No. 4, 511–536 (1977).

    Google Scholar 

  4. S. N. Naboko, “A functional model of perturbation theory and its applications to scattering theory,” Trudy Mat. Inst. Akad. Nauk SSSR,147, 86–114 (1980).

    Google Scholar 

  5. A. V. Kuzhel' and Yu. L. Kudryashov, “Symmetric and selfadjoint dilations of dissipative operators,” Dokl. Akad. Nauk SSSR,253, No. 4, 812–815 (1980).

    Google Scholar 

  6. Yu. L. Kudryashov, “Symmetric and selfadjoint dilations of dissipative operators,” Teor. Funktsii Funktsional. Anal. i Prilozhen. (Khar'kov), No. 37, 51–54 (1982).

    Google Scholar 

  7. A. V. Kuzhel', “Selfadjoint and J-selfadjoint dilations of linear operators,” Teor. Funktsii Funktsional. Anal. i Prilozhen. (Khar'kov), No. 37, 54–62 (1982).

    Google Scholar 

  8. M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, New York (1975).

    Google Scholar 

  9. B. Sz.-Nagy and C. Foiaş, Harmonic Analysis of Operators on Hubert Space, North-Holland, Amsterdam (1970).

    Google Scholar 

  10. A. V. Kuzhel', “Regular extensions of Hermitian operators,” Dokl. Akad. Nauk SSSR,251, No. 1, 30–33 (1980).

    Google Scholar 

  11. A. V. Shtraus, “Characteristic functions of linear operators,” Izv. Akad. Nauk SSSR, Ser. Mat.,24, No. 1, 43–74 (1960).

    Google Scholar 

  12. A. N. Kochubei, “On the extensions of symmetric operators and symmetric binary relations,” Mat. Zametki,17, No. 1, 41–48 (1975).

    Google Scholar 

  13. V. M. Bruk, “On extensions of symmetric relations,” Mat. Zametki,22, No. 6, 825–834 (1977).

    Google Scholar 

  14. V. A. Derkach and M. M. Malamud, “The Weyl function of a Hermitian operator and its relation with the characteristic function,” Preprint DonFTI-85-9(104), Donetsk (1985).

  15. N. K. Nikol'skii and S. V. Khrushchev, “A functional model and certain problems of the spectral theory of functions,” Trudy Mat. Inst. Akad. Nauk SSSR,176, 97–210 (1987).

    Google Scholar 

  16. A. V. Shtraus, “On extensions and the characteristic function of a symmetric operator,” Izv. Akad. Nauk SSSR, Ser. Mat.,32, No. 1, 186–207(1968).

    Google Scholar 

  17. M. S. Livshits, “On a class of linear operators in a Hilbert space,” Mat. Sb.,19, No. 2, 239–260 (1946).

    Google Scholar 

  18. Yu. M. Arlinskii, “On regular (*)-extensions and characteristic matrix functions of ordinary differential equations,” in: Boundary Value Problems for Differential Equations [in Russian], Naukova Dumka, Kiev (1980), pp. 3–13.

    Google Scholar 

  19. F. G. Maksudov and B. P. Allakhverdiev, “On the theory of the characteristic function and on the spectral analysis of a dissipative Schrödinger operator,” Dokl. Akad. Nauk SSSR,303, No. 6, 1307–1309 (1988).

    Google Scholar 

  20. E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Part I, Oxford University Press, Oxford (1962).

    Google Scholar 

Download references

Authors
  1. B. M. Solomyak
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 178, pp. 57–91, 1989.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Solomyak, B.M. Functional model for dissipative operators. A coordinate-free approach. J Math Sci 61, 1981–2002 (1992). https://doi.org/10.1007/BF01095663

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01095663

Keywords

Search

Navigation