Illinois Journal of Mathematics

Compact composition operators with symbol a universal covering map onto a multiply connected domain

Matthew M. Jones

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We generalise previous results of the author concerning the compactness of composition operators on the Hardy spaces $H^{p}$, $1\leq p<\infty$, whose symbol is a universal covering map from the unit disk in the complex plane to general finitely connected domains. We demonstrate that the angular derivative criterion for univalent symbols extends to this more general case. We further show that compactness in this setting is equivalent to compactness of the composition operator induced by a univalent mapping onto the interior of the outer boundary component of the multiply connected domain.

Article information

Source
Illinois J. Math., Volume 59, Number 3 (2015), 707-715.

Dates
Received: 6 July 2015
Revised: 27 April 2016
First available in Project Euclid: 30 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1475266405

Digital Object Identifier
doi:10.1215/ijm/1475266405

Mathematical Reviews number (MathSciNet)
MR3554230

Zentralblatt MATH identifier
1353.47052

Subjects
Primary: 47B33: Composition operators
Secondary: 30F35: Fuchsian groups and automorphic functions [See also 11Fxx, 20H10, 22E40, 32Gxx, 32Nxx]

Citation

Jones, Matthew M. Compact composition operators with symbol a universal covering map onto a multiply connected domain. Illinois J. Math. 59 (2015), no. 3, 707--715. doi:10.1215/ijm/1475266405. https://projecteuclid.org/euclid.ijm/1475266405


Export citation

References

  • A. F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, Springer-Verlag, New York, 1983.
  • C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions, CRC Press, Boca Raton, FL, 1995.
  • H. M. Farkas and I. Kra, Riemann surfaces, Graduate Texts in Mathematics, Springer-Verlag, New York, 1980.
  • J. B. Garnett and D. E. Marshall, Harmonic measure, Cambridge University Press, Cambridge, 2005.
  • M. M. Jones, Compact composition operators with symbol a universal covering map, J. Funct. Anal. 268 (2015), 887–901.
  • J. H. Shapiro, The essential norm of a composition operator, Ann. of Math. (2) 125 (1987), 375–404.
  • J. H. Shapiro, Composition operators and classical function theory, Springer-Verlag, New York, 1993.
  • J. H. Shapiro and P. D. Taylor, Compact, nuclear, and Hilbert–Schmidt composition operators on ${H}^2$, Indiana Univ. Math. J. 23 (1973), 471–496.