Skip to main content
SpringerLink
Log in

A factorization method for products of holomorphic matrix functions

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

A class of matrix functions defined on a contour which bounds a finitely connected domain in the complex plane is considered. It is assumed that each matrix function in this class can be explicitly represented as a product of two matrix functions holomorphic in the outer and the inner part of the contour, respectively. The problem of factoring matrix functions in the class under consideration is studied. A constructive method reducing the factorization problem to finitely many explicitly written systems of linear algebraic equations is proposed. In particular, explicit formulas for partial indices are obtained.

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.

Similar content being viewed by others

References

  1. I. I. Privalov, Boundary Properties of Analytic Functions (Gosudarstv. Izdat. Tehn. -Teor. Lit., Moscow–Leningrad, 1950) [in Russian].

    Google Scholar 

  2. G. Ts. Tumarkin and S. Ya. Khavinson, “Classes of analytic functions in multiply connected domains,” in Research on Modern Problems of the Theory of Functions of a Complex Variable (Fizmatgiz, Moscow, 1960), pp. 45–77 [in Russian].

    Google Scholar 

  3. G. S. Litvinchuk and I. M. Spitkovskii, Factorization of Measurable Matrix Functions, in Operator Theory: Advances and Applications (Birkhäuser Verlag, Basel, 1987), Vol. 25.

    Book  Google Scholar 

  4. F. D. Gakhov, “The Riemann boundary-value problem for a system of n pairs of functions,” Uspekhi Mat. Nauk 7 (4 (50)), 3–54 (1952).

    MATH  Google Scholar 

  5. I. M. Spitkovskii, “Factorization of measurable matrix functions. Related questions of the theory of systems of singular integral equations and the vector Riemann boundary value problem. I,” Differ. Uravn. 17 (4), 697–709 (1981).

    MathSciNet  Google Scholar 

  6. A. G. Kamalyan, “Some properties of kernels of Toeplitz operators,” Dokl. Nats. Akad. Nauk Armenii 107 (4), 316–322 (2007).

    MathSciNet  Google Scholar 

  7. A. G. Kamalyan, “Index factorization of matrix functions,” Dokl. Nats. Akad. Nauk Armenii 108 (1), 5–11 (2008).

    MathSciNet  Google Scholar 

  8. I. Gohberg, L. Lerer, and L. Rodman, “Factorization indices for matrix polynomials,” Bull. Amer. Math. Soc. 84 (2), 275–277 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  9. V. M. Adukov, “Wiener–Hopf factorization of meromorphic matrix functions,” Algebra i Analiz 4 (1), 54–74 (1992) [St. PetersburgMath. J. 4 (1), 51–69 (1993)].

    MathSciNet  MATH  Google Scholar 

  10. A. G. Kamalyan, “Explicit generalized factorization of bounded holomorphic matrix functions,” Izv. Nats. Akad. Nauk ArmeniiMat. 32 (2), 19–38 (1997) [J. Contemp. Math. Anal. 32(1997) (2), 14–32 (1998)].

    MathSciNet  MATH  Google Scholar 

  11. A. Böttcher and Yu. I. Karlovich, Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators, in Progress inMath. (Birkhäuser, Basel, 1997), Vol. 154.

    Book  MATH  Google Scholar 

  12. V. M. Adukov, “On exact and approximate solution of the Wiener–Hopf factorization problem for meromorphic matrix functions,” Vestn. Yuzhno-Ural. Gos. Univ. Ser. Mat. Fiz. Khim. 10 (7), 3–12 (2008).

    Google Scholar 

  13. V. M. Adukov, “Wiener–Hopf factorization of piecewise meromorphic matrix-valued functions,” Mat. Sb. 200 (8), 3–24 (2009) [Sb. Math. 200 (8), 1105–1126 (2009)].

    Article  MathSciNet  MATH  Google Scholar 

  14. R. A. Amirdzhanyan and A. G. Kamalyan, “Factorization of meromorphic matrix functions,” Izv. Nats. Akad. Nauk Armenii. Mat. 42 (6), 31–50 (2007) [J. Contemp. Math. Anal. 42 (6), 303–319 (2007)].

    MathSciNet  Google Scholar 

  15. A. G. Kamalyan and A. V. Sargsyan, “On partial indices of a class of second-order matrix functions,” Izv. Nats. Akad. Nauk Armenii. Mat. 42 (3), 39–48 (2007) [ J. Contemp. Math. Anal. 42 (3), 139–145 (2007)].

    MathSciNet  MATH  Google Scholar 

  16. A. G. Kamalyan and A. V. Sargsyan, “Solvability of some singular integral equations on the circle with shift,” Izv. Nats. Akad. Nauk Armenii. Mat. 46 (3), 29–46 (2011) [ J. Contemp. Math. Anal. 46 (3), 142–156 (2011)].

    MathSciNet  MATH  Google Scholar 

  17. A. G. Kamalyan, A. G. Stepanyan, and G. M. Topikyan, “Integral equations with a Hilbert kernel,” Izv. Nats. Akad. Nauk Armenii. Mat. 47 (2), 31–44 (2012) [ J. Contemp. Math. Anal. 47 (2), 51–61 (2012)].

    MathSciNet  MATH  Google Scholar 

  18. V. N. Gavdzinskii and I. M. Spitkovskii, “On a method for effectively constructing factorization,” Ukrain. Mat. Zh. 34 (1), 15–19 (1982).

    Article  MathSciNet  Google Scholar 

  19. A. B. Lebre, “Factorization in the Wiener algebra of a class of 2 × 2 matrix functions,” Integral Equations Operator Theory 12 (3), 408–423 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  20. I. Feldman, I. Gohberg, and N. Krupnik, “A method of explicit factorization of matrix functions and applications,” Integral Equations Operator Theory 18 (3), 277–302 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  21. I. Feldman, I. Gohberg, and N. Krupnik, “An explicit factorization algorithm,” Integral Equations Operator Theory 49 (2), 149–164 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  22. V. M. Adukov, “On classes ofmatrix functions admitting an explicit solution of theWiener–Hopf factorization problem,” Izv. Chelyabinsk. Nauchn. Centra 41 (3 (41)), 12–17 (2008).

    MathSciNet  Google Scholar 

  23. S. N. Kiyasov, “Some cases of effective factorization of second-order matrix functions,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 6, 36–43 (2012) [RussianMath. (Iz. VUZ) 56 (6), 30–36 (2012)].

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Yerevan State University, Yerevan, Armenia

    A. G. Kamalyan

  2. Institute of Mathematics, National Academy of Sciences of Armenia, Yerevan, Armenia

    A. G. Kamalyan

Authors
  1. A. G. Kamalyan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. G. Kamalyan.

Additional information

Original Russian Text © A. G. Kamalyan, 2016, published in Matematicheskie Zametki, 2016, Vol. 100, No. 2, pp. 212–228.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kamalyan, A.G. A factorization method for products of holomorphic matrix functions. Math Notes 100, 213–228 (2016). https://doi.org/10.1134/S0001434616070178

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434616070178

Keywords

Search

Navigation