Arkiv för Matematik

  • Ark. Mat.
  • Volume 37, Number 1 (1999), 171-182.

Norm convergence of normalized iterates and the growth of Kœnigs maps

Pietro Poggi-Corradini

Full-text: Open access

Abstract

Let ϕ be an analytic function defined on the unit disk D, with ϕ(D)⊂D, ϕ(0)=0, and ϕ′(0)=λ≠0. Then by a classical result of G. Kœnigs, the sequence of normalized iterates Φnn converges uniformly on compact subsets of D to a function σ analytic in D which satisfies σ°φ=λσ. It is of interest in the study of composition operators to know if, whenever σ belongs to a Hardy space Hp, the sequence Φnn converges to σ in the norm of Hp. We show that this is indeed the case, generalizing a result of P. Bourdon obtained under the assumption that ϕ is univalent.

When ϕ is inner, P. Bourdon and J. Shapiro have shown that σ does not belong to the Nevanlinna class, in particular it does not belong to any Hp. It is natural to ask, how bad can the growth of σ be in this case? As a partial answer we show that σ always belongs to some Bergman space L ${}_{a}^{p}$ .

Note

The author is partially supported by NSF Grant DMS 97-06408 and wishes to thank Professor P. Bourdon for sharing results and conjectures.

Article information

Source
Ark. Mat., Volume 37, Number 1 (1999), 171-182.

Dates
Received: 4 August 1997
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898621

Digital Object Identifier
doi:10.1007/BF02384832

Mathematical Reviews number (MathSciNet)
MR1673430

Zentralblatt MATH identifier
1029.30017

Rights
1999 © Institut Mittag-Leffler

Citation

Poggi-Corradini, Pietro. Norm convergence of normalized iterates and the growth of Kœnigs maps. Ark. Mat. 37 (1999), no. 1, 171--182. doi:10.1007/BF02384832. https://projecteuclid.org/euclid.afm/1485898621


Export citation

References

  • [B] Bourdon, P., Convergence of the Kœnigs sequence, in Studies on Composition Operators (Jafari, F., MacCluer, B. D., Cowen, C. C. and Porter, A. D., eds.), Contemp. Math. 213, pp. 1–10, Amer. Math. Soc., Providence, R. I., 1998.
  • [BS] Bourdon, P. and Shapiro, J., Mean growth of Kœnigs eigenfunctions, J. Amer. Math. Soc. 10 (1997), 299–325.
  • [CG] Carleson, L. and Gamelin, T., Complex Dynamics, Springer-Verlag, Berlin-Heidelberg-New York, 1993.
  • [D] Duren, P., Theory of HpSpaces, Academic Press, New York, 1970.
  • [G] Garnett, J., Bounded Analytic Functions, Academic Press, Orlando, Fla., 1981.
  • [K] Korenblum, B., An extension of the Nevanlinna theory, Acta Math., 135 (1975), 187–219.
  • [P1] Poggi-Corradini, P., The Hardy class of Kœnigs maps, Mich. Math. J. 44 (1997), 495–507.
  • [P2] Poggi-Corradini, P., The Hardy class of geometric models and the essential spectral radius of composition operators, J. Funct. Anal., 143 (1997), 129–156.
  • [R] Rudin, W., Function Theory in the Unit Ball ofCn, Springer-Verlag, Berlin-Heidelberg-New York, 1980.
  • [S] Shapiro, J., Composition Operators and Classical Function Theory, Springer-Verlag, Berlin-Heidelberg-New York, 1993.