Advertisement

Complex Analysis and Operator Theory

, Volume 13, Issue 8, pp 4027–4044 | Cite as

Closed Range Type Properties of Toeplitz Operators on the Bergman Space and the Berezin Transform

  • Nina ZorboskaEmail author
Article

Abstract

We characterize the multiplication operators with closed range on the Bergman space in terms of the Berezin transform, and apply this characterization to finite products of interpolating Blaschke products. We give some necessary and some sufficient conditions for invertibility of general Toeplitz operators on the Bergman space. We determine the Fredholm Toeplitz operators with \(BMO^1\) symbols and the invertible Toeplitz operators with nonnegative symbols, when their Berezin transform is bounded and of vanishing oscillation.

Keywords

Multiplication operator Toeplitz operator Bergman space Berezin transform Closed range operator Invertible operator Fredholm operator Interpolating Blaschke product 

Mathematics Subject Classification

47B35 47A53 30H20 30H35 30J10 

Notes

Funding

Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada (CA) (Grant No. 1304332).

References

  1. 1.
    Axler, S., Zheng, D.: Compact operators via the Berezin transform. Indiana Univ. Math. J. 47, 387–400 (1998)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Axler, S., Zheng, D.: The Berezin transform on the Toeplitz algebra. Studia Math. 127, 113–136 (1998)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Chicoń, K.: Closed range multiplication operators on weighted Bergman spaces. Nonlinear Anal. 60, 37–48 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Davidson, K., Douglas, R.: The generalized Berezin transform and commutator ideals. Pacific J. Math. 222, 29–56 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ghatage, P., Tjani, M.: Closed range composition operators on Hilbert function spaces. J. Math. Anal. Appl. 431, 841–866 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gorkin, P.: Functions not vanishing on trivial Gleason parts of Douglas algebras. Proc. Am. Math. Soc. 104, 1086–1090 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Luecking, D.: Inequalities on Bergman spaces. Illinois J. Math. 25, 1–11 (1981)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Luecking, D.: Characterization of certain classes of Hankel operators on the Bergman spaces. J. Funct. Anal. 110, 247–271 (1992)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Luecking, D.: Bounded composition operators with closed range on the Dirichlet space. Proc. Am. Math. Soc. 128, 1109–1116 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Marshal, D.E., Stray, A.: Interpolating Blaschke products. Pacific J. Math. 173, 491–499 (1996)MathSciNetCrossRefGoogle Scholar
  11. 11.
    McDonald, G., Sundberg, C.: Toeplitz operators on the disc. Indiana Univ. Math. J. 28, 595–611 (1979)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Nikolskii, N.K.: Treatise on the Shift Operator. Springer, Berlin (1986)CrossRefGoogle Scholar
  13. 13.
    Stroethoff, K., Zheng, D.: Toeplitz and Hankel operators on Bergman spaces. Trans. Am. Math. Soc. 329, 773–794 (1992)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Suarez, D.: The essential norm in the Toeplitz algebra on \(A^p (\mathbb{B}_n)\). Indiana Univ. Math. J. 56, 2185–2232 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Taskinen, J., Virtanen, J.: Toeplitz operators on Bergman spaces with locally integrable symbols. Rev. Math. Iberoamericana 26, 693–706 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Zhao, X., Zheng, D.: Invertibility of Toeplitz operators via Berezin Transforms. J. Oper. Theory 75, 475–495 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zhu, K.: VMO, ESV, and Toeplitz operators on the Bergman space. Trans. Am. Math. Soc. 302, 617–646 (1987)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zhu, K.: Operator Theory in Function Spaces. Marcel Dekker, New York (1990)zbMATHGoogle Scholar
  19. 19.
    Zorboska, N.: Toeplitz operators with BMO symbols and the Berezin transform. Int. J. Math. Sci. 46, 2929–2945 (2003)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ManitobaWinnipegCanada

Personalised recommendations