Skip to main content
SpringerLink
Log in

Boundary values of a convergent sequence of J-contractive matrix-functions

  • Published:
Mathematical notes of the Academy of Sciences of the USSR Aims and scope Submit manuscript

Abstract

In this note it is proved that if Wn(z) are J-contractive matrix-functions which are meromorphic in the disk ¦z¦<1 (J−W *n (z)JWn(z)≥0, J*=J, J2=I), Wn(z)→W(z) as n→∞,

$$W^* (z)JW(z) \leqslant W_n^* (z)JW_n (z)$$

and

$$\det W(z)\not \equiv 0,$$

then there exists a subsequence\(W_{n_k }\)(z) whose boundary values

$$W_{n_k }^* (\zeta )JW_{n_k } (\zeta ) \to W^* (\zeta )JW(\zeta )(\left| \zeta \right| = 1a.e.).$$

It follows from this result that every convergent Blaschke-Potapov product has J-unitary boundary values.

1. Let J be a fixed self-adjoint unitary matrix, i.e., J* = J and J2 = I. A matrix W is called J-contractive (J-unitary) ifJ−W * JW⩾0 (J−W * JW=0).

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. V. P. Potapov, “The multiplicative structure of J-contractive matrix-functions,” Trudy Moskovskogo Obshchestva,4, 125–236 (1955).

    Google Scholar 

  2. V. P. Potapov, “Basic facts of the theory of J-contractive analytic matrix-functions,” in: Theses of Reports at the All-Union Conference on the Theory of Functions of a Complex Variable [in Russian], Kharkov (1971), pp. 179–181.

  3. M. G. Krein and Sh. N. Saakyan, “On some new results in the theory of the resolvents of Hermitian operators,” Dokl. Akad. Nauk. SSSR,169, No. 6, 1269–1272 (1966).

    Google Scholar 

  4. M. G. Krein and Sh. N. Saakyan, “The resolvent matrix of a Hermitian operator and the characteristic functions connected with it,” Funktsional'. Analiz i Ego Prilozhen.,4, No. 3, 103–104 (1970).

    Google Scholar 

  5. M. S. Livshits, Operators, Oscillations, and Waves [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  6. M. S. Livshits and A. A. Yantsevich, Theory of Operator Nodes in Hubert Spaces [in Russian], Izd. Khar'kovskogo Universiteta, Kharkov (1971).

    Google Scholar 

  7. I. V. Kovalishina and V. P. Potapov, “An indefinite metric in the Nevanlinna-Pic Problem,” Dokl. Akad. Nauk ArmSSR,59, No. 1, 17–22 (1974).

    Google Scholar 

  8. I. V. Kovalishina, “J-noncontractive matrix-functions and the classical problems of analysis,” in: Theses of Reports at the All-Union Conference on the Theory of Functions of a Complex Variable [in Russian], Kharkov (1971), pp. 97–98.

  9. I. V. Kovalishina, “J-noncontractive matrix-functions in the Carathéodory problem,” Dokl. Akad. Nauk. ArmSSR,59, No. 3, 129–135 (1974).

    Google Scholar 

  10. I. V. Kovalishina, “J-noncontractive matrix functions and the classical problem of moments,” Dokl. Akad. Nauk. ArmSSR,60, No. 1, 3–10 (1975).

    Google Scholar 

  11. I. P. Fedchina, “A description of the solutions of the Nevanlinna-Pic tangent problem,” Dokl. Akad. Nauk. ArmSSR,60, No. 1, 37–42 (1975).

    Google Scholar 

  12. S. A. Orlov, “A theorem on the stabilization of nested matrix disks, analytically depending on a parameter,” in: Theses of Reports at the All Union Conference on the Theory of Functions of a Complex Variable [in Russian], Kharkov (1971), pp. 165–167.

  13. S. A. Orlov, “On the convergence and character of convergence of monotone families of J-contractive analytic matrix-functions,” Dokl. Akad. Nauk. ArmSSR,59, No. 4, 193–198 (1974).

    Google Scholar 

  14. S. A. Orlov, “Nested matrix disks analytically depending on a parameter and a theorem on the invariance of the ranks of radii of limiting matrix disks,” Izv. Akad. Nauk SSSR,40, No. 3 (1976).

  15. F. Atkinson, Discrete and Continuous Boundary-Value Problems, Academic Press, New York (1964).

    Google Scholar 

  16. A. V. Efimov and V. P. Potapov, “J-noncontractive matrix-functions and their role in the analytic theory of circuits,” Usp. Matem. Nauk,28, No. 1, 65–130 (1973).

    Google Scholar 

  17. I. I. Privalov, Boundary Properties of Analytic Functions [in Russian], Gostekhizdat, Moscow-Leningrad (1950).

    Google Scholar 

  18. Yu. P. Ginzburg, “On J-contractive operators in a Hubert space,” Nauchn. Zap. Odessk. Ped. Instituta,22, No. 1, 13–19 (1958).

    Google Scholar 

  19. R. Nevanlinna, “Über beschränkte analytische Funktionen,” Ann. Akad. Scient. Fenn., Ser. A,32, No. 7, 1–75 (1929).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Odessa Pedagogical Institute, USSR

    D. Z. Arov & L. A. Simakova

  2. Odessa Technological Institute for Refrigeration Industry, USSR

    D. Z. Arov & L. A. Simakova

Authors
  1. D. Z. Arov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. A. Simakova
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 491–500, April, 1976.

The authors are thankful to V. P. Potapov for useful remarks which made the preparation of this note possible.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Arov, D.Z., Simakova, L.A. Boundary values of a convergent sequence of J-contractive matrix-functions. Mathematical Notes of the Academy of Sciences of the USSR 19, 301–306 (1976). https://doi.org/10.1007/BF01156787

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01156787

Keywords

Search

Navigation