Journal of Mathematical Sciences

, Volume 234, Issue 1, pp 82–97 | Cite as

Factorization of generalized γ-generating matrices

  • Olena SukhorukovaEmail author


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.


γ-generating matrices J-inner matrix-valued function denominator associated pair generalized Schur class reproducing kernel space Potapov–Ginzburg transform Krein–Langer factorization 


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  1. 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. 2.
    V. M. Adamjan, “Nondegenerate unitary couplings of semiunitary operators,” Funkts. Anal. Prilozh., 7, No. 4, 1–16 (1973).MathSciNetGoogle Scholar
  3. 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.zbMATHGoogle Scholar
  4. 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).CrossRefzbMATHGoogle Scholar
  5. 5.
    N. Aronszajn, “Theory of reproducing kernels,” Trans. Amer. Math. Soc., 68, 337–404 (1950).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    D. Z. Arov, “Realization of matrix-valued functions according to Darlington,” Izv. Akad. Nauk SSSR. Ser. Mat., 37, 1299–1331 (1973).MathSciNetzbMATHGoogle Scholar
  7. 7.
    D. Z. Arov, “Regular and singular J-inner matrix functions and corresponding extrapolation problems,” Funct. Anal. Appl., 22, No. 1, 46–48 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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).Google Scholar
  9. 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).CrossRefGoogle Scholar
  10. 10.
    D. Z. Arov and H. Dym,J-Contractive Matrix-Valued Functions and Related Topics, Cambridge Univ. Press, Cambridge, 2008.CrossRefzbMATHGoogle Scholar
  11. 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.Google Scholar
  12. 12.
    J. Bognar, Indefinite Inner Product Spaces, Springer, Heidelberg, 1974.CrossRefzbMATHGoogle Scholar
  13. 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).Google Scholar
  14. 14.
    V. A. Derkach and O. Sukhorukova, “Generalized γ-generating matrices and Nehari–Takagi problem,” Oper. Matrices, 10, No. 4, 1073–1091 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 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).MathSciNetzbMATHGoogle Scholar
  16. 16.
    B. Francis, A Course in H Control Theory, Springer, Berlin, 1987.CrossRefGoogle Scholar
  17. 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. 18.
    L. Schwartz, “Sous-espaces hilbertiens d’espaces vectoriels topologiques et noyaux associes,” J. Analyse Math., 13, 115–256 (1964).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 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).MathSciNetzbMATHGoogle Scholar
  20. 20.
    O. Sukhorukova, “Generalized γ-generating matrices,” J. of Math. Sci., 218, No. 1, 89–104 (2016).MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dragomanov National Pedagogical UniversityKievUkraine

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