Skip to main content
SpringerLink
Log in

Self-Adjoint Toeplitz Operators Associated with Representing Measures on Multiply Connected Planar Regions and Their Eigenvalues

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

Abstract

A direct calculation of the vector of Riemann constants corresponding to the marked double of a multiply connected planar region is given. The existence of eigenvalues of self-adjoint Toeplitz operators acting on Hardy spaces associated with non-negative representing measures on 1-holed planar regions is established in the case where there exists one bounded component in the complement of the essential range of the symbol \(\phi \) of the operator. The analysis is done by using the zeros of translations of theta functions restricted to \(\mathbb R \) in \(\mathbb C \).

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Rosenblum, M.: A concrete spectral theory for self-adjoint Toeplitz operators. Am. J. Math. 87, 709–718 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  2. Rosenblum, M.: Self-adjoint Toeplitz operators and associated orthonormal functions. Proc. Am. Math. Soc. 13, 590–595 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  3. Akbari Estahbanati, G. (now known as Cyrus Parsi Aryana).: Riemann surfaces and Toeplitz operators on multiply connected planar regions. Dissertation, The University of Georgia (1993)

  4. Abrahamse, M.B.: Toeplitz operators in multiply connected regions. Am. J. Math. 96, 261–297 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  5. Aryana, C.P., Clancey, K.F.: On the existence of eigenvalues of Toeplitz operators on planar regions. Proc. Am. Math. Soc. 132(10), 3007–3018 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Akbari Estahbanati, G.: On the spectral character of Toeplitz operators on planar regions. Proc. Am. Math. Soc. 124(9), 2737–2744 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Clancey, K.F.: On the spectral character of Toeplitz operators on multiply connected domains. Trans. Am. Math. Soc. 323(2), 897–910 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ball, J.A., Clancey, K.F.: Reproducing kernels for hardy spaces on multiply connected domains. Integral Equ. Oper. Theory 25(1), 3551 (1996)

    Article  MathSciNet  Google Scholar 

  9. Fay, J.D.: Theta Functions on Riemann Surfaces. Lecture Notes in Mathematics, vol. 98. Springer-Verlag, New York (1973)

  10. Farkas, H.M., Kra, I.: Riemann Surfaces. Springer, New York (1992)

    Book  MATH  Google Scholar 

  11. Clancey, K.F.: Representing measures on multiply connected planar domains. Ill. J. Math. 35(2), 286–311 (1991)

    MathSciNet  MATH  Google Scholar 

  12. Mumford, D.: Tata Lectures on Theta II. Birkhäuser, Basel (1993)

    Google Scholar 

  13. Clancey, K.F.: Toeplitz operators on multiply connected domains and theta functions. Operator Theory Adv. Appl. 35, 311–355 (1988)

    MathSciNet  Google Scholar 

  14. David, N.: Representing measures and topological type of finite bordered Riemann surfaces. Trans. Am. Math. Soc. 192, 120–138 (1974)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Saginaw Valley State University, University Center, MI, 48710, USA

    Cyrus Parsi Aryana

Authors
  1. Cyrus Parsi Aryana
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cyrus Parsi Aryana.

Additional information

Communicated by Joseph Ball.

A portion of this paper forms a portion of the author’s doctoral dissertation [3]

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aryana, C.P. Self-Adjoint Toeplitz Operators Associated with Representing Measures on Multiply Connected Planar Regions and Their Eigenvalues. Complex Anal. Oper. Theory 7, 1513–1524 (2013). https://doi.org/10.1007/s11785-012-0261-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11785-012-0261-7

Keywords

Mathematics Subject Classification (2000)

Search

Navigation