Advertisement

Complex Analysis and Operator Theory

, Volume 13, Issue 1, pp 151–192 | Cite as

\(C^*\)-algebras of Bergman Type Operators with Piecewise Continuous Coefficients Over Domains with Dini-Smooth Corners

  • Enrique Espinoza-Loyola
  • Yuri I. KarlovichEmail author
Article
  • 40 Downloads

Abstract

For any simply connected domain \(U\subset \mathbb {C}\) with a piecewise Dini-smooth boundary \(\partial U\) which admits Dini-smooth corners of openings lying in \((0,2\pi ]\), the \(C^*\)-algebra \({\mathfrak B}_U=\mathrm{alg}\{aI,B_U,\widetilde{B}_U:a\in PC({\mathfrak L})\}\) generated by the operators of multiplication by piecewise continuous functions with discontinuities on a finite union \({\mathfrak L}\subset U\) of piecewise Dini-smooth curves that have one-sided tangents at every point, do not form cusps and are not tangent to \(\partial U\) at the points of \({\mathfrak L}\cap \partial U\), and by the Bergman projection \(B_U\) and anti-Bergman projection \(\widetilde{B}_U\) acting on the Lebesgue space \(L^2(U)\) is studied. A Fredholm symbol calculus for the \(C^*\)-algebra \({\mathfrak B}_U\) is constructed and a Fredholm criterion for the operators \(A\in {\mathfrak B}_U\) in terms of their symbols is established. Results essentially depend on openings of corners and angles between curves of discontinuity of coefficients at the points \(z\in \partial U\cap {\mathfrak L}\).

Keywords

Bergman and anti-Bergman projections Piecewise continuous function Dini-smooth corner \(C^*\)-algebra Fredholm symbol calculus Fredholmness 

Mathematics Subject Classification

Primary 47L15 Secondary 47A53 47G10 47L30 

Notes

Acknowledgements

The authors are grateful to the referees for the useful comments and suggestions.

References

  1. 1.
    Ahlfors, L.V.: Lectures on Quasiconformal Mappings. D. Van Nostrand, Princeton (1966)zbMATHGoogle Scholar
  2. 2.
    Böttcher, A., Karlovich, Y.I.: Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators. Progress in Mathematics, vol. 154. Birkhäuser, Basel (1997)CrossRefzbMATHGoogle Scholar
  3. 3.
    Böttcher, A., Silbermann, B.: Analysis of Toeplitz Operators, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  4. 4.
    Dzhuraev, A.: Methods of Singular Integral Equations. Longman Scientific & Technical, Harlow (1992)zbMATHGoogle Scholar
  5. 5.
    Douglas, R.G.: Banach Algebra Techniques in Operator Theory. Academic Press, New York (1972)zbMATHGoogle Scholar
  6. 6.
    Espinoza-Loyola, E., Karlovich, Y.I.: \(C^*\)-algebras of Bergman type operators with piecewise continuous coefficients over bounded polygonal domains. In: Maz’ya, V., Natroshvili, D., Shargorodsky, E., Wendland, W.L. (eds.) Recent Trends in Operator Theory and Partial Differential Equations. The Roland Duduchava Anniversary Volume, Operator Theory: Advances and Applications, vol. 258, pp. 145–171. Birkhäuser, Basel (2017)CrossRefGoogle Scholar
  7. 7.
    Espinoza-Loyola, E., Karlovich, Y.I., Vilchis-Torres, O.: \(C^*\)-algebras of Bergman type operators with piecewise constant coefficients over sectors. Integr. Equ. Oper. Theory 83, 243–269 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Karlovich, Y.I.: \(C^*\)-algebras of Bergman type operators with continuous coefficients on polygonal domains. Oper. Matrices 9, 773–802 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Karlovich, Y.I.: \(C^*\)-algebras of poly-Bergman type operators over sectors. Integr. Equ. Oper. Theory 87, 435–460 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Karlovich, Y.I., Pessoa, L.: Algebras generated by Bergman and anti-Bergman projections and by multiplications by piecewise continuous coefficients. Integr. Equ. Oper. Theory 52, 219–270 (2005)CrossRefzbMATHGoogle Scholar
  11. 11.
    Karlovich, Y.I., Pessoa, L.V.: \(C^*\)-algebras of Bergman type operators with piecewise continuous coefficients. Integr. Equ. Oper. Theory 57, 521–565 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Karlovich, Y.I., Pessoa, L.V.: Poly-Bergman projections and orthogonal decompositions of \(L^2\)-spaces over bounded domains. In: Bastos, M.A., Gohberg, I., Lebre, A.B., Speck, F.-O. (eds.) Operator Algebras, Operator Theory and Applications. Operator Theory: Advances and Applications, vol. 181, pp. 263–282. Birkhäuser, Basel (2008)CrossRefGoogle Scholar
  13. 13.
    Karlovich, Y.I., Pessoa, L.V.: \(C^*\)-algebras of Bergman type operators with piecewise continuous coefficients on bounded domains. In: Begehr, H.G.W., Nicolosi, F. (eds.) More Progresses in Analysis (Proceedings of the 5th International ISAAC Congress, Catania, Italy, 25–30 July 2005), pp. 339–348. World Scientific, Singapore (2009)Google Scholar
  14. 14.
    Loaiza, M.: Algebras generated by the Bergman projection and operators of multiplication by piecewise continuous functions. Integr. Equ. Oper. Theory 46, 215–234 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Loaiza, M.: On the algebra generated by the harmonic Bergman projection and operators of multiplication by piecewise continuous functions. Bol. Soc. Mat. Mexicana III(10), 179–193 (2004)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Loaiza, M.: On an algebra of Toeplitz operators with piecewise continuous symbols. Integr. Equ. Oper. Theory 51, 141–153 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mikhlin, S.G.: Multidimensional Singular Integrals and Integral Equations. Pergamon Press, Oxford (1965)zbMATHGoogle Scholar
  18. 18.
    Mikhlin, S.G., Prössdorf, S.: Singular Integral Operators. Springer, Berlin (1986)CrossRefzbMATHGoogle Scholar
  19. 19.
    Pessoa, L.V.: The method of variation of the domain for poly-Bergman spaces. Math. Nachr. 286, 1850–1862 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Pommerenke, C.: Boundary Behaviour of Conformal Maps. Springer, Berlin (1992)CrossRefzbMATHGoogle Scholar
  21. 21.
    Rabinovich, V.S., Roch, S., Silbermann, B.: Limit Operators and Their Applications in Operator Theory. Operator Theory: Advances and Applications, vol. 150. Birkhäuser, Basel (2004)CrossRefzbMATHGoogle Scholar
  22. 22.
    Roch, S., Santos, P.A., Silbermann, B.: Non-commutative Gelfand Theories. A Tool-Kit for Operator Theorists and Numerical Analysts. Springer, London (2011)zbMATHGoogle Scholar
  23. 23.
    Simonenko, I.B.: Chin Ngok Min: Local Method in the Theory of One-Dimensional Singular Integral Equations with Piecewise Continuous Coefficients. University Press, Rostov on Don, Noetherity (1986). (Russian)Google Scholar
  24. 24.
    Timan, A.F.: Theory of Approximation of Functions of a Real Variable. Pergamon Press, Oxford (1963)zbMATHGoogle Scholar
  25. 25.
    Vasilevski, N.L.: Banach algebras generated by two-dimensional integral operators with a Bergman kernel and piecewise continuous coefficients. I. Soviet Math. (Izv. VUZ) 30(2), 14–24 (1986)MathSciNetGoogle Scholar
  26. 26.
    Vasilevski, N.L.: Toeplitz operators on the Bergman spaces: inside-the domain effects. Contemp. Math. 289, 79–146 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Vasilevski, N.L.: Commutative Algebras of Toeplitz Operators on the Bergman Space. Birkhäuser, Basel (2008)zbMATHGoogle Scholar
  28. 28.
    Zhu, K.: Operator Theory in Function Spaces. Marcel Dekker, New York (1990)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Instituto de Matemáticas, Unidad CuernavacaUniversidad Nacional Autónoma de MéxicoCuernavacaMexico
  2. 2.Centro de Investigación en Ciencias, Instituto de Investigación en Ciencias Básicas y AplicadasUniversidad Autónoma del Estado de MorelosCuernavacaMexico

Personalised recommendations