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On Connection Between Factorizations of Weighted Schur Functions and Invariant Subspaces

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Operator Theory, Analysis and Mathematical Physics

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 174))

Abstract

We study operator-valued functions of weighted Schur classes over multiply-connected domains. There is a correspondence between functions of weighted Schur classes and so-called “conservative curved” systems introduced in the paper. In the unit disk case the fundamental relationship between invariant subspaces of the main operator of a conservative system and factorizations of the corresponding Schur class function (characteristic function) is well known. We extend this connection to weighted Schur classes. With this aim we develop new notions and constructions and make suitable changes in standard theory.

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References

  1. B. Szökefalvi-Nagy, C. Foiaş, Harmonic analysis of operators on Hilbert space. North-Holland, Amsterdam-London, 1970.

    Google Scholar 

  2. M.S. Brodskii, Unitary operator colligations and their characteristic functions, Uspekhi mat. nauk 33 (1978), no. 4, 141–168.

    MathSciNet  Google Scholar 

  3. P.L. Duren, Theory of H p spaces, Pure Appl. Math., vol. 38, Academic Press, New York-London, 1970.

    Google Scholar 

  4. N.K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1 Hardy, Hankel, and Toeplitz. Vol. 2 Model operators and systems, Math. Surveys and Monographs, 92, 93, AMS, Providence, RI, 2002.

    Google Scholar 

  5. M.S. Livsic, On a class of linear operators on Hilbert space, Mat. Sb. 19 (1946), 239–260.

    Google Scholar 

  6. A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1949), 239–255.

    Article  Google Scholar 

  7. H. Bart, I. Gohberg, M.A. Kaashoek, Minimal factorization of matrix and operator functions. Operator Theory: Adv. and Appl., 1979.

    Google Scholar 

  8. A.S. Tikhonov, Functional model and duality of spectral components for operators with continuous spectrum on a curve. Algebra i Analiz, 14 (2002), no. 4, 158–195.

    MATH  MathSciNet  Google Scholar 

  9. A.S. Tikhonov, Free functional model related to simply-connected domains. Operator Theory: Adv. and Appl., Vol. 154 (2004), 405–415.

    MathSciNet  Google Scholar 

  10. M.B. Abrahamse, R.G. Douglas, A class of subnormal operators related to multiply connected domains, Adv. in Math., 19 (1976), 106–148.

    Article  MATH  MathSciNet  Google Scholar 

  11. J.A. Ball, Operators of class C 00 over multiply connected domains, Michigan Math. J., 25 (1978), 183–195.

    Article  MATH  MathSciNet  Google Scholar 

  12. B.S. Pavlov, S.I. Fedorov, Group of shifts and harmonic analysis on Riemann surface of genus one, Algebra i Analiz, 1 (1989), no. 2, 132–168.

    MATH  MathSciNet  Google Scholar 

  13. S.I. Fedorov, On harmonic analysis in multiply connected domain and character-automorphic Hardy spaces. Algebra i Analiz, 9 (1997), no. 2, 192–239.

    Google Scholar 

  14. D.V. Yakubovich, Linearly similar model of Sz.-Nagy-Foias type in a domain. Algebra i Analiz, 15 (2003), no. 2, 180–227.

    Google Scholar 

  15. S.M. Verduyn Lunel, D.V. Yakubovich, A functional model approach to linear neutral functional differential equations, Integr. Equ. Oper. Theory, 27 (1997), 347–378.

    Article  MATH  MathSciNet  Google Scholar 

  16. N.K. Nikolski, V.I. Vasyunin, Elements of spectral theory in terms of the free functional model. Part I: Basic constructions, Holomorphic spaces (eds. Sh. Axler, J. Mc-Carthy, D. Sarason), MSRI Publications 33 (1998), 211–302.

    Google Scholar 

  17. A.S. Tikhonov, Transfer functions for “curved” conservative systems. Operator Theory: Adv. and Appl., Vol. 153 (2004), 255–264.

    Article  MathSciNet  Google Scholar 

  18. S.S. Boiko, V.K. Dubovoi, Unitary couplings and regular factorizations of operator functions in L∞, Dopovidi Nacional. Akad. Nauk Ukr., (1997), no. 1, 41–44.

    MathSciNet  Google Scholar 

  19. A.S. Tikhonov, Inner-outer factorization of J-contractive-valued functions. Operator Theory: Adv. and Appl., Vol. 118 (2000), 405–415.

    MathSciNet  Google Scholar 

  20. W. Mlak, Intertwining operators. Studia Math., Vol. 43 (1972), 219–233.

    MATH  MathSciNet  Google Scholar 

  21. L. Kerchy, On the factorization of operator-valued functions Acta. Sci. Math. (Szeged), 69 (2003), 337–348.

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Taurida National University, 4, Yaltinskaya str., 95007, Simferopol, Crimea, Ukraine

    Alexey Tikhonov

Authors
  1. Alexey Tikhonov
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Editor information

Editors and Affiliations

  1. Institut of Mathematics, Polish Academy of Sciences, ul. Sw. Tomasza 30/7, 31-027, Krakow, Poland

    Jan Janas

  2. Department of Mathematics, Lund Institute of Technology, 221 00, Lund, Sweden

    Pavel Kurasov

  3. Dept. of Mathematics, Imperial College London, Room 675 South Kensington Campus, SW7 2AZ, London, UK

    Ari Laptev

  4. Department of Mathematics, Royal Institute of Technology, 100 44, Stockholm, Sweden

    Ari Laptev

  5. Department of Mathematical Physics, St. Petersburg State University, Ulyanovskaya 1, 198904, St. Petersburg, Russia

    Sergei Naboko

  6. Department of Mathematics, University of Alabama, 452 Campbell Hall, Birmingham, AL, 35294-1170, USA

    Günter Stolz

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© 2007 Birkhäuser Verlag Basel/Switzerland

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Tikhonov, A. (2007). On Connection Between Factorizations of Weighted Schur Functions and Invariant Subspaces. In: Janas, J., Kurasov, P., Laptev, A., Naboko, S., Stolz, G. (eds) Operator Theory, Analysis and Mathematical Physics. Operator Theory: Advances and Applications, vol 174. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8135-6_13

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