Hardy spaces and twisted sectors for geometric models
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- by Pietro Poggi-Corradini PDF
- Trans. Amer. Math. Soc. 348 (1996), 2503-2518 Request permission
Abstract:
We study the one-to-one analytic maps $\sigma$ that send the unit disc into a region $G$ with the property that $\lambda G\subset G$ for some complex number $\lambda$, $0<|\lambda |<1$. These functions arise in iteration theory, giving a model for the self-maps of the unit disk into itself, and in the study of composition operators as their eigenfunctions. We show that for such functions there are geometrical conditions on the image region $G$ that characterize their rate of growth, i.e. we prove that $\sigma \in \bigcap _{p<\infty }H^p$ if and only if $G$ does not contain a twisted sector. Then, we examine the connection with composition operators, and further investigate the no twisted sector condition. Finally, in the Appendix, we give a different proof of a result of J. Shapiro about the essential norm of a composition operator.References
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Additional Information
- Pietro Poggi-Corradini
- Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195-4350
- Email: poggi@math.washington.edu
- Received by editor(s): November 16, 1994
- Received by editor(s) in revised form: June 13, 1995
- Additional Notes: The author acknowledges support from INDAM (Istituto Nazionale di Alta Matematica) while studying at the University of Washington, and wishes to thank Professor D. Marshall for his help and advice.
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 2503-2518
- MSC (1991): Primary 30C45, 30D55, 47B38
- DOI: https://doi.org/10.1090/S0002-9947-96-01564-4
- MathSciNet review: 1340184