Skip to main content
SpringerLink
Log in

Interpolating Sequences in Harmonically Weighted Dirichlet Spaces

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

In this article we show that interpolating sequences on certain harmonically weighted Dirichlet spaces can be characterized in terms of a separation condition and a Carleson-measure condition. This is the first example of a space with Nevanlinna–Pick kernel with non-radially symmetric weights in which this characterization remains true.

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. Agler, J., McCarthy, J.: Pick interpolation and Hilbert function spaces. Graduate Studies in Mathematics, vol. 44. American Mathematical Society, Providence, RI (2002)

  2. Aleman, A.: The Multiplication Operators on Hilbert Spaces of Analytic Functions. Habilitationsschrift, Fernuniversität Hagen (1993)

  3. Bishop, C.: Interpolating sequences for the Dirichlet space and its multipliers, Preprint (1994)

  4. Boe B.: Interpolating sequences for Besov spaces. J. Funct. Anal. 192, 319–341 (2002)

    Article  MathSciNet  Google Scholar 

  5. Boe, B.: An interpolation Theorem for Hilbert spaces with Nevanlinna–Pick kernel. Proc. Am. Math. Soc. 210 (2003)

  6. Chartrand R.: Toeplitz operators on Dirichlet-type spaces. J. Oper. Theory 48, 3–13 (2002)

    MATH  MathSciNet  Google Scholar 

  7. Chartrand R.: Multipliers and Carleson measures for D(μ). Integral Equ. Oper. Theory 45, 309–318 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. de Branges L., Rovnyak J.: Square Summable Power Series. Holt, Rinehart, and Winston, New York (1966)

    MATH  Google Scholar 

  9. Marshall, D.E., Sundberg, C.: Interpolating sequences for the multipliers of the Dirichlet space, Preprint (1994). Available at http://www.math.washington.edu/~marshall/preprints/preprints.html

  10. McCullough S., Trent T.: Invariant subspaces and Nevanlinna–Pick kernels. J. Funct. Anal. 178, 226–249 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Richter S.: A representation theorem for cyclic analytic two-isometries. Trans. Am. Math. Soc. 328, 325–349 (1991)

    Article  MATH  Google Scholar 

  12. Richter S., Sundberg C.: A formula for the local Dirichlet integral. Michigan Math. J. 38, 355–379 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  13. Richter S., Sundberg C.: Multipliers and invariant subspaces in the Dirichlet space. J. Oper. Theory 28, 167–187 (1992)

    MATH  MathSciNet  Google Scholar 

  14. Sarason D.: Local Dirichlet spaces as De Branges–Rovnyak spaces. Proc. Am. Math. Soc. 125, 2133–2139 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. Sarason D.: Harmonically weighted Dirichlet spaces associated with finitely atomic measures. Integral Equ. Oper. Theory 31, 186–213 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Seip, K.: Interpolation and Sampling in Spaces of Analytic Functions. University Lecture Series, vol. 33. American Mathematical Society, Providence, RI (2004)

  17. Serra A.: Interpolating sequences in harmonically weighted Dirichlet spaces. Proc. Am. Math. Soc. 131, 2809–2817 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Shimorin S.: Reproducing kernels and extremal functions in Dirichlet-type spaces. J. Math. Sci. 107, 4108–4124 (2001)

    Article  MathSciNet  Google Scholar 

  19. Shimorin S.: Complete Nevanlinna–Pick property of Dirichlet type spaces. J. Funct. Anal. 191, 276–296 (2002)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Medición y Evaluación, Universidad de los Andes, Mérida, 5101, Venezuela

    Gerardo R. Chacón

Authors
  1. Gerardo R. Chacón
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gerardo R. Chacón.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chacón, G.R. Interpolating Sequences in Harmonically Weighted Dirichlet Spaces. Integr. Equ. Oper. Theory 69, 73–85 (2011). https://doi.org/10.1007/s00020-010-1820-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00020-010-1820-9

Mathematics Subject Classification (2010)

Keywords

Search

Navigation