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Composition operators on weighted Bergman spaces of a half-plane

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

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.

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References
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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).
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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.
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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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