Angular derivatives at boundary fixed points

for self-maps of the disk

Author:
Pietro Poggi-Corradini

Journal:
Proc. Amer. Math. Soc. **126** (1998), 1697-1708

MSC (1991):
Primary 30D05, 30D55, 30C35, 47B38, 58F23

MathSciNet review:
1443404

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a one-to-one analytic function of the unit disk into itself, with . The origin is an attracting fixed point for , if is not a rotation. In addition, there can be fixed points on where has a finite angular derivative. These boundary fixed points must be repelling (abbreviated b.r.f.p.). The Koenigs function of is a one-to-one analytic function defined on such that , where . If is the first iterate of that does have b.r.f.p., we compute the Hardy number of , , in terms of the smallest angular derivative of at its b.r.f.p.. In the case when no iterate of has b.r.f.p., then , and vice versa. This has applications to composition operators, since is a formal eigenfunction of the operator . When acts on , by a result of C. Cowen and B. MacCluer, the spectrum of is determined by and the essential spectral radius of , . Also, by a result of P. Bourdon and J. Shapiro, and our earlier work, can be computed in terms of . Hence, our result implies that the spectrum of is determined by the derivative of at the fixed point and the angular derivatives at b.r.f.p. of or some iterate of .

**1.**Carl C. Cowen and Barbara D. MacCluer,*Spectra of some composition operators*, J. Funct. Anal.**125**(1994), no. 1, 223–251. MR**1297020**, 10.1006/jfan.1994.1123**2.**C. Cowen and B. MacCluer,*Composition operators on spaces of analytic functions*, CRC Press, 1995. CMP**96:14****3.**Peter L. Duren,*Theory of 𝐻^{𝑝} spaces*, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR**0268655****4.**Herbert Kamowitz,*The spectra of composition operators on 𝐻^{𝑝}.*, J. Functional Analysis**18**(1975), 132–150. MR**0407645****5.**D. Marshall,*Angular derivatives and Lipschitz majorants*, preprint.**6.**Pietro Poggi-Corradini,*Hardy spaces and twisted sectors for geometric models*, Trans. Amer. Math. Soc.**348**(1996), no. 6, 2503–2518. MR**1340184**, 10.1090/S0002-9947-96-01564-4**7.**P. Poggi-Corradini,*The Hardy class of geometric models and the essential spectral radius of composition operators*, J. Functional Analysis, 143, (1997), 129-156. CMP**97:06****8.**Ch. Pommerenke,*Boundary behaviour of conformal maps*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR**1217706****9.**Joel H. Shapiro,*Composition operators and classical function theory*, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR**1237406****10.**Joel H. Shapiro, Wayne Smith, and David A. Stegenga,*Geometric models and compactness of composition operators*, J. Funct. Anal.**127**(1995), no. 1, 21–62. MR**1308616**, 10.1006/jfan.1995.1002

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Additional Information

**Pietro Poggi-Corradini**

Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903

Email:
pp2n@virginia.edu

DOI:
https://doi.org/10.1090/S0002-9939-98-04303-2

Received by editor(s):
January 18, 1996

Received by editor(s) in revised form:
November 15, 1996

Additional Notes:
The author was supported by the University of Washington Math. Department while at MSRI, Berkeley, in the Fall of 1995. He also wishes to thank Professor D. Marshall for his help and advice.

Communicated by:
Theodore W. Gamelin

Article copyright:
© Copyright 1998
American Mathematical Society