Skip to main content
SpringerLink
Log in

Compact linear combinations of composition operators induced by linear fractional maps

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

It has been known that the difference of two composition operators induced by linear fractional self-maps of a ball cannot be nontrivially compact on either the Hardy space or any standard weighted Bergman space. In this paper we extend this result in two significant directions: the difference is extended to general linear combinations and inducing maps are extended to linear fractional maps taking a ball into another possibly of different dimension.

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. Choe, B.R., Koo, H., Park, I.: Compact differences of composition operators over polydisks. Integral Equ. Oper. Theory 73(1), 57–91 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Choe, B.R., Koo, H., Park, I.: Compact differences of composition operators on the Bergman spaces over the ball. Potential Anal. 40, 81–102 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cowen, C.C., MacCluer, B.D.: Composition Operators on Spaces of Analytic Fuctions, Studies in Advanced Mathematics. CRC Press, Boca Raton (1995)

    Google Scholar 

  4. Cowen, C.C., MacCluer, B.D.: Linear fractional maps of the ball and their composition operators. Acta Sci. Math. (Szeged) 66(1–2), 351–376 (2000)

    MathSciNet  MATH  Google Scholar 

  5. Gallardo-Gutiérrez, E., González, M., Nieminen, P., Saksman, E.: On the connected component of compact composition operators on the Hardy space. Adv. Math. 219, 986–1001 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Heller, K., MacCluer, B.D., Weir, R.J.: Compact differences of composition operators in several variables. Integral Equ. Oper. Theory 69(2), 247–268 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hunziker, H., Jarchow, H., Mascioni, V.: Some topologies on the space of analytic self-maps of the unit disk, Geometry of Banach spaces (Strobl, 1989), London Math. Soc. Lecture Note Ser. 158, pp. 133–148, Cambridge Univ. Press, Cambridge (1990)

  8. Jiang, L., Ouyang, C.: Compact differences of composition operators on holomorphic function spaces in the unit ball. Acta Math. Sci. Ser. B Engl. Ed. 31(5), 1679–1693 (2011)

    MathSciNet  MATH  Google Scholar 

  9. Koo, H., Smith, W.: Composition operators induced by smooth self-maps of the unit ball in \({\mathbf{C}}^{N}\). J. Math. Anal. Appl. 329, 617–633 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Koo, H., Wang, M.: Joint Carleson measure and the difference of composition operators on \(A^p_\alpha (\mathbf{B}_n)\). J. Math. Anal. Appl. 419, 1119–1142 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kriete, T., Moorhouse, J.: Linear relations in the Calkin algebra for composition operators. Trans. Am. Math. Soc. 359(6), 2915–2944 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. MacCluer, B.D.: Components in the space of composition operators. Integral Equ. Oper. Theory 12(5), 725–738 (1989)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Saukko, E.: An application of atomic decomposition in Bergman spaces to the study of differences of composition operators. J. Funct. Anal. 262, 3872–3890 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Shapiro, J.H.: Composition Operators and Classical Function Theory. Springer, New York (1993)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball. Springer, New York (2005)

    MATH  Google Scholar 

Download references

Acknowledgments

Part of this research was performed during the third author’s visit to Korea University in January 2014. He thanks the mathematics department of Korea University for their hospitality and support.

Author information

Authors and Affiliations

  1. Department of Mathematics, Korea University, Seoul, 136-713, Korea

    Boo Rim Choe, Hyungwoon Koo & Jongho Yang

  2. School of Mathematics and Statistics, Wuhan University, Wuhan, China

    Maofa Wang

Authors
  1. Boo Rim Choe
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hyungwoon Koo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Maofa Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jongho Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jongho Yang.

Additional information

B. R. Choe was supported by NRF (2013R1A1A2004736) of Korea, H. Koo was supported by NRF (2012R1A1A2000705) of Korea and NSFC (11271293), and M. Wang was supported by NSFC (11271293).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Choe, B.R., Koo, H., Wang, M. et al. Compact linear combinations of composition operators induced by linear fractional maps. Math. Z. 280, 807–824 (2015). https://doi.org/10.1007/s00209-015-1449-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-015-1449-0

Keywords

Mathematics Subject Classification

Search

Navigation