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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Hardy spaces and twisted sectors
for geometric models


Author: Pietro Poggi-Corradini
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
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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 [Enhancements On Off] (What's this?)

  • 1. L. Ahlfors, Conformal Invariants, McGraw-Hill, 1973. MR 50:10211
  • 2. L. Carleson, et al. (eds.), The collected works of Arne Beurling, Vol. 1, Birkhäuser, 1989. MR 92k:01046a
  • 3. P. Duren, Theory of $H^p$ spaces, Academic Press, 1970.
  • 4. G. Königs, Recherches sur les intégrales de certaines équations functionnelles, Annales Ecole Normale Supérieure (3) 1 (1884), Supplément, 3-41.
  • 5. J. Garnett, and D. Marshall, Harmonic measure, to appear.
  • 6. L. Hansen, Hardy classes and ranges of functions, Michigan Math. J. 17 (1970), 235-248. MR 41:7118
  • 7. P. Poggi-Corradini, The Hardy class of geometric models and the essential spectral radius of composition operators, preprint.
  • 8. C. Pommerenke, Schlichte Funktionen und analytische Funktionen von beschränkter mittler Ozillation, Comm. Math. Helv. 52 (1977), 591-602. MR 56:11268
  • 9. J. H. Shapiro, W. Smith, and D. Stegenga, Geometric models and compactness of composition operators, J. Funct. Anal. 127 (1995), 21--62. CMP 95:05
  • 10. J. H. Shapiro and P. D. Taylor, Compact, nuclear, and Hilbert-Schmidt composition operators on $H^2$, Indiana Univ. Math. J. 23 (1973/74), 471-496. MR 48:4816
  • 11. J. H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, 1993. MR 94k:47049
  • 12. J. H. Shapiro, The essential norm of a composition operator, Annals of Math. 125 (1987), 375-404. MR 88c:47058
  • 13. D. Stegenga and K. Stephenson, A geometric characterization of analytic functions with bounded mean oscillation, J. London Math. Soc. (2) 24 (1981), 243-254. MR 82m:30036

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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

DOI: https://doi.org/10.1090/S0002-9947-96-01564-4
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.
Article copyright: © Copyright 1996 American Mathematical Society

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