Mean growth of Koenigs eigenfunctions

Authors:
Paul S. Bourdon and Joel H. Shapiro

Journal:
J. Amer. Math. Soc. **10** (1997), 299-325

MSC (1991):
Primary 30D05, 47B38

DOI:
https://doi.org/10.1090/S0894-0347-97-00224-5

MathSciNet review:
1401457

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In 1884, G. Koenigs solved Schroeder’s functional equation \begin{equation*} f\circ \phi = \lambda f \end{equation*} in the following context: $\phi$ is a given holomorphic function mapping the open unit disk $U$ into itself and fixing a point $a\in U$, $f$ is holomorphic on $U$, and $\lambda$ is a complex scalar. Koenigs showed that if $0 < |\phi ’(a)| < 1$, then Schroeder’s equation for $\phi$ has a unique holomorphic solution $\sigma$ satisfying \begin{equation*} \sigma \circ \phi = \phi ’(a) \sigma \qquad \text {and}\qquad \sigma ’(0) = 1; \end{equation*} moreover, he showed that the only other solutions are the obvious ones given by constant multiples of powers of $\sigma$. We call $\sigma$ the Koenigs eigenfunction of $\phi$. Motivated by fundamental issues in operator theory and function theory, we seek to understand the growth of integral means of Koenigs eigenfunctions. For $0 < p < \infty$, we prove a sufficient condition for the Koenigs eigenfunction of $\phi$ to belong to the Hardy space $H^p$ and show that the condition is necessary when $\phi$ is analytic on the closed disk. For many mappings $\phi$ the condition may be expressed as a relationship between $\phi ’(a)$ and derivatives of $\phi$ at points on $\partial U$ that are fixed by some iterate of $\phi$. Our work depends upon a formula we establish for the essential spectral radius of any composition operator on the Hardy space $H^p$.

- I. N. Baker and Ch. Pommerenke,
*On the iteration of analytic functions in a half-plane II*, J. London Math. Soc. (2) 20 (1979), 255–258. - P. S. Bourdon and J. H. Shapiro,
*Riesz composition operators*, preprint. - S. R. Caradus, W. E. Pfaffenberger, and B. Yood,
*Calkin algebras and algebras of operators on Banach spaces*, Marcel Dekker, New York, 1974. - J. Caughran and H. J. Schwartz,
*Spectra of compact composition operators*, Proc. Amer. Math. Soc. 51 (1975), 127–130. - C. C. Cowen,
*Iteration and the solution of functional equations for functions analytic in the unit disk*, Trans. Amer. Math. Soc. 265 (1981), 69–95. - C. C. Cowen,
*Composition operators on $H^2$*, J. Operator Th. 9 (1983),77–106. - C. C. Cowen and B. D. MacCluer,
*Composition Operators on Spaces of Analytic Functions*, CRC Press, Boca Raton, 1995. - C. C. Cowen and B. D. MacCluer,
*Spectra of some composition operators*, J. Funct. Anal. 125 (1994), 223–251. - P. L. Duren,
*Theory of $H^p$ Spaces*, Academic Press, New York, 1983. - M. Essén, D.F. Shea, and C.S. Stanton,
*A value-distribution criterion for the class $L$ Log $L$, and some related questions*, Ann. Inst. Fourier (Grenoble) 35 (1985), 127–150. - B. Gramsch,
*Integration und holomorphe Funktionen in lokalbeschränkten Räumen*, Math. Annalen 162 (1965), 190–210. - L. J. Hanson,
*Hardy classes and ranges of functions*, Mich. Math. J. 17 (1970), 235–248. - W. K. Hayman,
*Multivalent Functions*, Cambridge Tracts in Mathematics #100, second ed., Cambridge University Press, 1994. - H. Kamowitz,
*The spectra of composition operators on $H^p$*, J. Funct. Anal. 18 (1975), 132–150. - G. Koenigs,
*Recherches sur les intégrales de certaines équationes functionelles*, Annales Ecole Normale Superior (3) 1 (1884), Supplément, 3–41. - B. D. MacCluer and J. H. Shapiro,
*Angular derivatives and compact composition operators on Hardy and Bergman spaces*, Canadian J. Math. 38 (1986), 878–906. - E. A. Nordgren,
*Composition operators*, Canadian J. Math. 20 (1968), 442–449. - P. Poggi-Corradini,
*Hardy spaces and twisted sectors for geometric models*, Trans. Amer. Math. Soc., to appear. - P. Poggi-Corradini,
*The Hardy class of geometric models and the essential spectral radius of composition operators*, preprint. - P. Poggi-Corradini,
*The Hardy class of Koenigs maps*, preprint. - Ch. Pommerenke,
*On the iteration of analytic functions in a half-plane I*, J. London Math Soc. (2) 19 (1979), 439–447. - W. Rudin,
*Real and Complex Analysis*, 3rd edition, McGraw Hill, New York, 1987. - J. V. Ryff,
*Subordinate $H^p$ functions*, Duke Math. J. 33 (1966), 347–354. - H. Schwartz,
*Composition operators on $H^p$*, Thesis: University of Toledo, 1969. - J. H. Shapiro,
*The essential norm of a composition operator*, Annals of Math. 125 (1987), 375–404. - J. H. Shapiro,
*Composition Operators and Classical Function Theory*, Springer-Verlag, New York, 1993. - J. H. Shapiro,
*The Riesz and Fredholm Theories in Linear Topological Spaces*, preprint. - J. H. Shapiro, W. Smith, and D. A. Stegenga,
*Geometric models and compactness of composition operators*, J. Funct. Anal. 127 (1995), 21–62. - G. Valiron,
*Sur l’iteration des fonctions holomorphes dans un demi-plan*, Bull des Sci. Math. (2) 55 (1931), 105–128.

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (1991):
30D05,
47B38

Retrieve articles in all journals with MSC (1991): 30D05, 47B38

Additional Information

**Paul S. Bourdon**

Affiliation:
Department of Mathematics, Washington and Lee University, Lexington, Virginia 24450

Email:
pbourdon@wlu.edu

**Joel H. Shapiro**

Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

Email:
shapiro@math.msu.edu

Received by editor(s):
January 22, 1996

Received by editor(s) in revised form:
June 19, 1996

Additional Notes:
The first author was supported in part by NSF grant DMS-9401206.

The second author was supported in part by NSF grant DMS-9424417

Article copyright:
© Copyright 1997
American Mathematical Society