Skip to main content
SpringerLink
Log in

Compact differences of composition operators on weighted Dirichlet spaces

  • Research Article
  • Published:
Central European Journal of Mathematics

Abstract

Here we consider when the difference of two composition operators is compact on the weighted Dirichlet spaces . Specifically we study differences of composition operators on the Dirichlet space and S 2, the space of analytic functions whose first derivative is in H 2, and then use Calderón’s complex interpolation to extend the results to the general weighted Dirichlet spaces. As a corollary we consider composition operators induced by linear fractional self-maps of the disk.

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. Barnes B.A., Continuity properties of the spectrum of operators on Lebesgue spaces, Proc. Amer. Math. Soc., 1989, 106(2), 415–421

    Article  MATH  MathSciNet  Google Scholar 

  2. Barnes B.A., Interpolation of spectrum of bounded operators on Lebesgue spaces, Rocky Mountain J. Math., 1990, 20(2), 359–378

    Article  MATH  MathSciNet  Google Scholar 

  3. Bourdon P.S., Components of linear-fractional composition operators, J. Math. Anal. Appl., 2003, 279(1), 228–245

    Article  MATH  MathSciNet  Google Scholar 

  4. Boyd D.M., Composition operators on the Bergman space, Colloq. Math., 1975/76, 34(1), 127–136

    MATH  MathSciNet  Google Scholar 

  5. Calderón A.-P., Intermediate spaces and interpolation, the complex method, Stud. Math., 1964, 24, 113–190

    MATH  Google Scholar 

  6. Conway J.B., A Course in Functional Analysis, 2nd ed., Grad. Texts in Math., 96, Springer, New York, 1990

    MATH  Google Scholar 

  7. Cowen C.C., MacCluer B.D., Composition Operators on Spaces of Analytic Functions, Stud. Adv. Math., CRC Press, Boca Raton, 1995

    Google Scholar 

  8. Cwikel M., Real and complex interpolation and extrapolation of compact operators, Duke Math. J., 1992, 65(2), 333–343

    Article  MATH  MathSciNet  Google Scholar 

  9. Heller K.C., Composition Operators on \(S^2 \left( \mathbb{D} \right)\), PhD thesis, University of Virginia, Charlottesville, 2010

    Google Scholar 

  10. Herrero D.A., Saxe-Webb K., Spectral continuity in complex interpolation, Math. Balkanica (N.S.), 1989, 3(3–4), 325–336

    MATH  MathSciNet  Google Scholar 

  11. MacCluer B.D., Components in the space of composition operators, Integral Equations Operator Theory, 1989, 12(5), 725–738

    Article  MATH  MathSciNet  Google Scholar 

  12. MacCluer B.D., Shapiro J.H., Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math., 1989, 38(4), 878–906

    MathSciNet  Google Scholar 

  13. McCarthy J.E., Geometric interpolation between Hilbert spaces, Ark. Mat., 1992, 30(2), 321–330

    Article  MATH  MathSciNet  Google Scholar 

  14. Moorhouse J., Compact differences of composition operators, J. Funct. Anal., 2005, 219(1), 70–92

    Article  MATH  MathSciNet  Google Scholar 

  15. Nordgren E.A., Composition operators, Canad. J. Math., 1968, 20, 442–449

    Article  MATH  MathSciNet  Google Scholar 

  16. Pons M.A., The spectrum of a composition operator and Calderón’s complex interpolation, In: Topics in Operator Theory, 1, Oper. Theory Adv. Appl., 202, Birkhäuser, Basel, 2010, 451–467

    Google Scholar 

  17. Saxe K., Compactness-like operator properties preserved by complex interpolation, Ark. Mat., 1997, 35(2), 353–362

    Article  MATH  MathSciNet  Google Scholar 

  18. Shapiro J.H., Compact composition operators on spaces of boundary-regular holomorphic functions, Proc. Amer. Math. Soc., 1987, 100(1), 49–57

    Article  MATH  MathSciNet  Google Scholar 

  19. Shapiro J.H., The essential norm of a composition operator, Ann. Math., 1987, 125(2), 375–404

    Article  MATH  Google Scholar 

  20. Shapiro J.H., Composition Operators and Classical Function Theory, Universitext Tracts Math., Springer, New York, 1993

    Book  Google Scholar 

  21. Shapiro J.H., Taylor P.D., Compact, nuclear, and Hilbert-Schmidt composition operators on H 2, Indiana Univ. Math. J., 1973/74, 23, 471–496

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin at La Crosse, 1725 State Street, La Crosse, WI, 54601, USA

    Robert F. Allen

  2. Department of Mathematics, North Central College, 30 N. Brainard St., Naperville, IL, 60540, USA

    Katherine C. Heller & Matthew A. Pons

Authors
  1. Robert F. Allen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Katherine C. Heller
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Matthew A. Pons
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert F. Allen.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Allen, R.F., Heller, K.C. & Pons, M.A. Compact differences of composition operators on weighted Dirichlet spaces. centr.eur.j.math. 12, 1040–1051 (2014). https://doi.org/10.2478/s11533-013-0397-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s11533-013-0397-3

MSC

Keywords

Search

Navigation