Skip to main content
SpringerLink
Log in

Criterion of unitary similarity of minimal passive scattering systems with a given transfer function

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We establish necessary and sufficient conditions under which all minimal passive scattering systems that have a given transfer operator function are unitarily equivalent. These conditions can be significantly simplified in special cases important for applications, in particular, in the case where a transfer function is rational and in a more general case where this function is pseudoextendable.

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

    MATH  MathSciNet  Google Scholar 

  2. R. E. Kalman, P. L. Falb, and M. A. Arbib, Topics in Mathematical System Theory, McGraw-Hill, New York 1969.

    MATH  Google Scholar 

  3. D. Z. Arov, Linear Stationary Passive Systems with Losses [in Russian], Doctoral-Degree Thesis (Physics and Mathematics), Odessa (1985).

  4. D. Z. Arov, “Stable dissipative linear stationary dynamic scattering systems,” J. Oper. Theory, 2, 95–126 (1979).

    MATH  MathSciNet  Google Scholar 

  5. D. Z. Arov, M. A. Kaashoek, and D. R. Ric. “Minimal and optimal linear discrete time-invariant dissipative scattering systems,” Integr. Equal. Oper. Theory, 29, 127–154 (1997).

    Article  MATH  Google Scholar 

  6. M. A. Nudel’man, “Optimal passive systems and semiboundedness of quadratic functionals,” Sib. Mat. Zh., 33, No. 1, 78–86 (1992).

    MathSciNet  Google Scholar 

  7. B. Szökefalvi-Nagy and C. Foias, Harmonic Analysis of Operators in Hilbert Spaces [Russian translation] Mir, Moscow 1970.

    Google Scholar 

  8. S. S. Boiko and V. K. Dubovoi, “On some extremal problems connected with the suboperator of scattering through inner channels of a system,” Dop. Nats. Akad. Nauk Ukr., No. 4, 7–11 (1997).

  9. M. S. Brodskii, “Unitary colligations and their characteristic functions,” Usp. Mat. Nauk, 33, No. 4 (202), 141–168 (1978).

  10. R. K. Mohamed, Defect Functions and Regular Extensions of Holomorphic Contracting Matrix Functions [in Russian], Candidate-Degree Thesis (Physics and Mathematics), Kharkov (1996).

  11. V. K. Dubovoi, B. Fritzsche, and B. Kirstein, “On spectrally associated Schur functions, Arov inner functions, and Nehari-type completion problem for Schur functions,” Integr. Equal. Oper. Theory, 17, 247–276 (1993).

    Article  MathSciNet  Google Scholar 

  12. V. K. Dubovoi and R. K. Mohamed, “Defect functions of holomorphic contracting functions, regular extensions, and open systems,” Math. Nachr., 160, 69–110 (1993).

    Article  MathSciNet  Google Scholar 

  13. V. M. Adamyan and D. Z. Arov, “On unitary couplings of semiunitary operators,” Mat. Issled., 1, No. 2, 3–66 (1966).

    MathSciNet  Google Scholar 

  14. Yu. A. Rozanov, Stationary Random Processes [in Russian] Fizmatgiz, Moscow 1963.

    Google Scholar 

  15. I.I. Privalov, Boundary Values of Analytic Functions [in Russian] Gostekhteoretizdat, Moscow-Leningrad 1950.

    Google Scholar 

  16. D. Z. Arov and M. A. Nudel’man, “Passive linear stationary dynamical scattering systems with continuous time,” Integr. Equat. Oper. Theory, 24, No. 1, 1–45 (1996).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Authors
  1. D. Z. Arov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. A. Nudel’man
    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

Arov, D.Z., Nudel’man, M.A. Criterion of unitary similarity of minimal passive scattering systems with a given transfer function. Ukr Math J 52, 161–172 (2000). https://doi.org/10.1007/BF02529631

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02529631

Keywords

Search

Navigation