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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
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.

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  • 1. Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0357743
  • 2. Arne Beurling, The collected works of Arne Beurling. Vol. 1, Contemporary Mathematicians, Birkhäuser Boston, Inc., Boston, MA, 1989. Complex analysis; Edited by L. Carleson, P. Malliavin, J. Neuberger and J. Wermer. MR 1057613
  • 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. Lowell J. Hansen, Hardy classes and ranges of functions, Michigan Math. J. 17 (1970), 235–248. MR 0262512
  • 7. P. Poggi-Corradini, The Hardy class of geometric models and the essential spectral radius of composition operators, preprint.
  • 8. Ernest S. Armstrong, On the linear optimal digital servo problem, Internat. J. Control 20 (1974), 347–348. MR 0452999
  • 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 𝐻², Indiana Univ. Math. J. 23 (1973/74), 471–496. MR 0326472
  • 11. Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1237406
  • 12. Joel H. Shapiro, The essential norm of a composition operator, Ann. of Math. (2) 125 (1987), no. 2, 375–404. MR 881273, 10.2307/1971314
  • 13. David A. Stegenga and Kenneth Stephenson, A geometric characterization of analytic functions with bounded mean oscillation, J. London Math. Soc. (2) 24 (1981), no. 2, 243–254. MR 631937, 10.1112/jlms/s2-24.2.243

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Additional Information

Pietro Poggi-Corradini
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195-4350

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