Skip to main content
SpringerLink
Log in

Factorization of generalized γ-generating matrices

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

The class of γ-generating matrices and its subclasses of regular and singular γ-generating matrices were introduced by D. Z. Arov in [8], where it was shown that every γ-generating matrix admits an essentially unique regular–singular factorization. The class of generalized γ-generating matrices was introduced in [14, 20]. In the present paper, subclasses of singular and regular generalized –generating matrices are introduced and studied. As the main result, a theorem of existence of the regular–singular factorization for a rational generalized γ-generating matrix is 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. V. M. Adamjan, D. Z. Arov, and M. G. Krein, “Infinite Hankel matrices and generalized problems of Caratheodory–Fejer and I. Schur problems,” Funkts. Anal. Prilozh., 2, No. 4, 1–17 (1968).

    Google Scholar 

  2. V. M. Adamjan, “Nondegenerate unitary couplings of semiunitary operators,” Funkts. Anal. Prilozh., 7, No. 4, 1–16 (1973).

    MathSciNet  Google Scholar 

  3. D. Alpay, A. Dijksma, J. Rovnyak, and H. S. V. de Snoo, Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces, \( {\mathrm{Birkh}}^{\begin{array}{l}{}^{\prime}\\ {}{}^{'}\end{array}}\mathrm{auser} \), Basel, 1997.

    MATH  Google Scholar 

  4. D. Alpay and H. Dym, “On applications of reproducing kernel spaces to the Schur algorithm and rational J unitary factorization. I. Schur methods in operator theory and signal processing,” Oper. Theory Adv. Appl., 18, 89–159 (1986).

    Article  MATH  Google Scholar 

  5. N. Aronszajn, “Theory of reproducing kernels,” Trans. Amer. Math. Soc., 68, 337–404 (1950).

    Article  MathSciNet  MATH  Google Scholar 

  6. D. Z. Arov, “Realization of matrix-valued functions according to Darlington,” Izv. Akad. Nauk SSSR. Ser. Mat., 37, 1299–1331 (1973).

    MathSciNet  MATH  Google Scholar 

  7. D. Z. Arov, “Regular and singular J-inner matrix functions and corresponding extrapolation problems,” Funct. Anal. Appl., 22, No. 1, 46–48 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  8. D. Z. Arov, “γ-generating matrices, j-inner matrix-functions and related extrapolation problems,” J. Soviet Math., 52, 3487–3491 (1990); 52, 3421–3425 (1990).

  9. D. Z. Arov and H. Dym, “J-inner matrix function, interpolation and inverse problems for canonical system, I: Foundation, integral equations,” Operator Theory, 28, 1–16 (1997).

    Article  Google Scholar 

  10. D. Z. Arov and H. Dym,J-Contractive Matrix-Valued Functions and Related Topics, Cambridge Univ. Press, Cambridge, 2008.

    Book  MATH  Google Scholar 

  11. T. Ya. Azizov and I. S. Iokhvidov, Foundations of the Theory of Linear Operators in Spaces with an Indefinite Metric [in Russian], Nauka, Moscow, 1986.

  12. J. Bognar, Indefinite Inner Product Spaces, Springer, Heidelberg, 1974.

    Book  MATH  Google Scholar 

  13. V. A. Derkach adn H. Dym, “On linear fractional transformations associated with generalized J-inner matrix functions,” Integ. Eq. Oper. Th., 65, 1–50 (2009).

  14. V. A. Derkach and O. Sukhorukova, “Generalized γ-generating matrices and Nehari–Takagi problem,” Oper. Matrices, 10, No. 4, 1073–1091 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  15. V. Derkach and O. Sukhorukova, “A-regular–A-singular factorizations of generalized J-inner matrix functions,” J. of Math. Sci., 23, No. 3, 231–251 (2017).

    MathSciNet  MATH  Google Scholar 

  16. B. Francis, A Course in H ∞ Control Theory, Springer, Berlin, 1987.

    Book  Google Scholar 

  17. M. G. Krein and H. Langer, “Über die verallgemeinerten Resolventen und die characteristische Function eines isometrischen Operators im Raume Πk,” Colloq. Math. Soc. Janos Bolyai, 5, 353–399 (1972).

    Google Scholar 

  18. L. Schwartz, “Sous-espaces hilbertiens d’espaces vectoriels topologiques et noyaux associes,” J. Analyse Math., 13, 115–256 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  19. O. Sukhorukova, “Factorization formulas for some classes of generalized J-inner matrix-valued functions,” Meth. Funct. Anal. Topology, 20, No. 4, 365–378 (2014).

    MathSciNet  MATH  Google Scholar 

  20. O. Sukhorukova, “Generalized γ-generating matrices,” J. of Math. Sci., 218, No. 1, 89–104 (2016).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dragomanov National Pedagogical University, Kiev, Ukraine

    Olena Sukhorukova

Authors
  1. Olena Sukhorukova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Olena Sukhorukova.

Additional information

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 14, No. 4, pp. 575–594 October–December, 2017.

This work was supported by a Volkswagen Stiftung grant and grant of the Ministry of Education and Science of Ukraine (project 0115U000556).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sukhorukova, O. Factorization of generalized γ-generating matrices. J Math Sci 234, 82–97 (2018). https://doi.org/10.1007/s10958-018-3983-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-018-3983-x

Keywords

Search

Navigation