Skip to main content
SpringerLink
Log in

Essential Norm of Weighted Composition Operators and Difference of Composition Operators Between Standard Weighted Bergman Spaces

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

We obtain estimates for the essential norm of weighted composition operators and the difference of composition operators between standard weighted Bergman spaces \(A^p_\alpha \). In particular, we generalize a result of J. Moorhouse which completely characterizes compactness of differences of composition operators on \(A^p_\alpha \).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Axler, S., Zheng, D.: Compact operators via the Berezin transform. Indiana Univ. Math. J. 47, 387–400 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bonet, J., Lindström, M., Wolf, E.: Differences of composition operators between weighted Banach spaces of holomorphic functions. J. Aust. Math. Soc. 84, 9–20 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Charpentier, S.: Essential norm of composition operators in the Hardy space \(H^1\) and the weighted Bergman space \(A^p_\alpha \) on the ball. Arch. Math. 98, 327–340 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cowen, C., MacCluer, B.: Composition operators on spaces of analytic functions. CRC Press, Boca Raton (1995)

    MATH  Google Scholar 

  5. Čučković, Z., Zhao, R.: Weighted composition operators on the Bergman space. J. Lond. Math. Soc. 70(2), 499–511 (2004)

    Article  MATH  Google Scholar 

  6. Čučković, Z., Zhao, R.: Weighted composition operators between different weighted Bergman spaces and different Hardy spaces. Ill. J. Math. 69 51, 479–498 (2007)

    MATH  Google Scholar 

  7. Čučković, Z., Zhao, R.: Essential norm estimates of weighted composition operators between Bergman spaces on strongly pseudoconvex domains. Math. Proc. Camb. Phil. Soc. 142, 525–533 (2007)

    Article  MATH  Google Scholar 

  8. Domanski, P., Lindström, M.: Sets of interpolation and sampling for weighted Banach spaces of holomorphic functions. Ann. Polon. Math. 79, 233–264 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Duren, P., Schuster, A.: Bergman spaces, AMS (2004)

  10. Galindo, P., Gamelin, T.W., Lindström, M.: Composition operators on uniform algebras, essential norms and hyperbolically bounded sets. Trans. Am. Math. Soc. 359, 2109–2121 (2007)

    Article  MATH  Google Scholar 

  11. Galindo, P., Laitila, J., Lindström, M.: The essential norm of a composition operator on BMOA. J. Funct. Anal. 265, 629–643 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hyvärinen, O., Kemppainen, M., Lindström, M., Rautio, A., Saukko, E.: The essential norms of weighted composition operators on weighted Banach spaces of analytic functions. Integral Equ. Oper. Theory 72, 151–157 (2012)

    Article  MATH  Google Scholar 

  13. Hyvärinen, O., Lindström, M.: Estimates of essential norms of weighted composition operators between Bloch type spaces. J. Math. Anal. Appl. 393, 38–44 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kriete, T., MacCluer, B.: Composition operators on large weighted Bergman spaces. Indiana Univ. Math. J. 41, 755–788 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lindström, M., Wolf, E.: Essential norm of the difference of weighted composition operators. Monatsh. Math. 153, 133–143 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Luecking, D.H.: Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives. Am. J. Math. 107, 85–111 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  17. MacCluer, B., Shapiro, J.: Angular derivatives and compact composition operators on the Hardy and Bergman spaces. Can. J. Math. 38, 878–906 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  18. MacCluer, B., Ohno, S., Zhao, R.: Topological structure of the space of composition operators on \(H^\infty \). Integral Equ. Oper. Theory 40, 481–494 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Manhas, J.S., Zhao, R.: New estimates of essential norms of weighted composition operators between Bloch type spaces. J. Math. Anal. Appl. 389, 32–47 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Moorhouse, J.: Compact differences of composition operators. J. Funct. Anal. 219, 70–92 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Moorhouse, J., Toews, C.: Differences of composition operators. Trends in Banach spaces and operator theory, pp. 207–213. Contemp. Math., 321, Amer. Math. Soc., Providence (2003)

  22. Nieminen, P.J.: Compact differences of composition operators on Bloch and Lipschitz spaces. Comput. Methods Function Theory 7, 325–344 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Nieminen, P.J., Saksman, E.: On compactness of the difference of composition operators. J. Math. Anal. Appl. 298, 501–522 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Pérez-González, F., Rättyä, J., Vukotić, D.: On composition operators acting between Hardy and weighted Bergman spaces. Expo. Math. 25(4), 309–323 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  25. Saukko, E.: Difference of composition operators between standard weighted Bergman spaces. J. Math. Anal. Appl. 381, 789–798 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  26. Shapiro, J.: The essential norm of a composition operator. Ann. Math. 125, 375–404 (1987)

    Article  MATH  Google Scholar 

  27. Shapiro, J., Sundberg, C.: Isolation amongst the composition operators. Pacific J. Math. 145, 117–152 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  28. Ueki, S.I.: Compact composition operators acting between weighted Bergman spaces of the unit ball. Arch. Math 93, 461–473 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  29. Ueki, S.I.: Compact weighted composition operators on weighted Bergman spaces. Acta Sci. Math. (Szeged) 75(3–4), 693–706 (2009)

    MATH  MathSciNet  Google Scholar 

  30. Zhao, R.: Essential norms of composition operators between Bloch type spaces. Proc. Am. Math. Soc. 138, 2537–2546 (2010)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Åbo Akademi University, FI-20500, Åbo, Finland

    Mikael Lindström

  2. Department of Mathematical Sciences, University of Oulu, FI-90014, Oulu, Finland

    Erno Saukko

Authors
  1. Mikael Lindström
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Erno Saukko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikael Lindström.

Additional information

Communicated by H.Turgay Kaptanoglu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lindström, M., Saukko, E. Essential Norm of Weighted Composition Operators and Difference of Composition Operators Between Standard Weighted Bergman Spaces. Complex Anal. Oper. Theory 9, 1411–1432 (2015). https://doi.org/10.1007/s11785-014-0430-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11785-014-0430-y

Keywords

Mathematics Subject Classification

Search

Navigation