Skip to main content

Composition operators on weighted Bergman spaces of a half-plane

  • Sam J. Elliott (a1) and Andrew Wynn (a2)

We use induction and interpolation techniques to prove that a composition operator induced by a map ϕ is bounded on the weighted Bergman space of the right half-plane if and only if ϕ fixes the point at ∞ non-tangentially and if it has a finite angular derivative λ there. We further prove that in this case the norm, the essential norm and the spectral radius of the operator are all equal and are given by λ(2+α)/2.

Hide All
1.Agler, J. and McCarthy, J. E., Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, Volume 44 (American Mathematical Society, Providence, RI, 2002).
2.Bergh, J. and Löfström, J., Interpolation spaces (Springer, 1976).
3.Cowen, C. C. and MacCluer, B. D., Composition operators on spaces of analytic functions, Studies in Advanced Mathematics (CRC Press, Boca Raton, FL, 1995).
4.Duren, P., Gallardo-Gutiérrez, E. A. and Montes-Rodríguez, A., A Paley–Wiener theorem for Bergman spaces with application to invariant subspaces, Bull. Lond. Math. Soc. 39 (2007), 459466.
5.Elliott, S. J. and Jury, M. T., Composition operators on Hardy spaces of a half plane, Bull. Lond. Math. Soc., in press.
6.Gallardo-Gutiérrez, E. A., Partington, J. R. and Segura, D., Cyclic vectors and invariant subspaces for Bergman and Dirichlet shifts, J. Operat. Theory 62 (2009), 199214.
7.Hedenmalm, H., Korenblum, B. and Zhu, K., Theory of Bergman spaces, Graduate Texts in Mathematics, Volume 199 (Springer, 2000).
8.Matache, V., Composition operators on Hardy spaces of a half-plane, Proc. Am. Math. Soc. 127 (1999), 14831491.
9.Matache, V., Weighted composition operators on H2 and applications, Complex Analysis Operat. Theory 2 (2008), 169197.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed