Advertisement

Ukrainian Mathematical Journal

, 63:658 | Cite as

On the fredholm theory of a planar problem with shift for a pair of functions

  • Z. M. Lysenko
  • L. V. Matviyuk
  • A. P. Nechaev
  • V. T. Shvets
Brief Communications
  • 25 Downloads

We establish necessary and sufficient conditions for the Fredholm property of a planar problem with shift and conjugation for a pair of functions and obtain a formula for the determination of its index.

Keywords

Planar Problem Toeplitz Operator Singular Integral Equation Quasiconformal Mapping Bergman Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Yu. I. Karlovich and L. Pessoa, “Algebras generated by Bergman and anti-Bergman projections and multiplications by piecewise continuous functions,” Integr. Equat. Oper. Theory, 52, 219–270 (2005).MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    N. L. Vasilevski, Commutative Algebras of Toeplitz Operators on the Bergman Space, Birkhäuser, Basel (2008).zbMATHGoogle Scholar
  3. 3.
    G. S. Litvinchuk, Boundary-Value Problems and Singular Integral Equations with Shift [in Russian], Nauka, Moscow (1977).Google Scholar
  4. 4.
    S. F. Skorokhod, Noether Theory of Many-Element Boundary-Value Problems with Shift for Functions Analytic in a Domain [in Russian], Candidate-Degree Thesis (Physics and Mathematics), Odessa (1984).Google Scholar
  5. 5.
    N. I. Lisovets, Investigation of Some Mixed Boundary-Value Problems in the Theory of Analytic Functions [in Russian], Candidate-Degree Thesis (Physics and Mathematics), Odessa (1984).Google Scholar
  6. 6.
    G. S. Litvinchuk, “On the operator approach in the theory of boundary-value problems with shift for functions analytic in a domain,” in: Proceedings of the Jubilee Seminar Dedicated to the 75th Birthday of F. D. Gakhov, an Academician of the Belorussian Academy of Sciences, [in Russian], Universitetskoe, Minsk (1985), pp. 69–76.Google Scholar
  7. 7.
    A. D. Dzhuraev, “On the theory of systems of singular integral equations on a bounded domain,” Dokl. Akad. Nauk SSSR, 249, No. 1, 22–25 (1979).MathSciNetGoogle Scholar
  8. 8.
    A. D. Dzhuraev, “Theory of some systems of singular integral equations over two-dimensional domains,” Dokl. Akad. Nauk SSSR, 279, No. 3, 528–532 (1984).MathSciNetGoogle Scholar
  9. 9.
    I. I. Komyak, “A class of two-dimensional singular integral operators in a circular domain,” Dokl. Akad. Nauk Belorus. SSR, 23, No. 11, 972–975 (1979).MathSciNetzbMATHGoogle Scholar
  10. 10.
    I. I. Komyak, “A class of two-dimensional singular integral equations with Bergman kernel,” Dokl. Akad. Nauk Belorus. SSR, 23, No. 1, 8–11 (1979).MathSciNetzbMATHGoogle Scholar
  11. 11.
    I. N. Vekua, Generalized Analytic Functions [in Russian], Nauka, Moscow (1988).Google Scholar
  12. 12.
    I. Kra, Automorphic Forms and Kleinian Groups, Benjamin, Massachusetts (1972).zbMATHGoogle Scholar
  13. 13.
    M. Schiffer and D. C. Spencer, Functionals on Finite Riemann Surfaces, Princeton University, Princeton (1954).Google Scholar
  14. 14.
    L. V. Ahlfors, Lectures on Quasiconformal Mappings, van Nostrand, Toronto (1966).zbMATHGoogle Scholar
  15. 15.
    J. R. Ortega, N. L. Vasilevski, and E. R. de Arellano, “On the algebra generated by the Bergman projection and shift operator. I,” Integr. Equat. Oper. Theory, 46, No. 4, pp. 455–471 (2003).zbMATHCrossRefGoogle Scholar
  16. 16.
    N. Ya. Krupnik, Banach Algebras with Symbol and Singular Integral Operators [in Russian], Shtiintsa, Kishinev (1984).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • Z. M. Lysenko
    • 1
  • L. V. Matviyuk
    • 1
  • A. P. Nechaev
    • 2
  • V. T. Shvets
    • 2
  1. 1.Mechnikov Odessa National UniversityOdessaUkraine
  2. 2.Odessa State Academy of RefrigerationOdessaUkraine

Personalised recommendations