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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
MathSciNet review: 1401457
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Abstract: In 1884, G. Koenigs solved Schroeder's functional equation

\begin{equation*}f\circ \varphi = \lambda f \end{equation*}

in the following context: $\varphi $ 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 $0 < |\varphi '(a)| < 1$, then Schroeder's equation for $\varphi $ has a unique holomorphic solution $\sigma $ satisfying

\begin{equation*}\sigma \circ\varphi = \varphi '(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 $\varphi $. 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 $\varphi $ to belong to the Hardy space $H^p$ and show that the condition is necessary when $\varphi $ is analytic on the closed disk. For many mappings $\varphi $ the condition may be expressed as a relationship between $\varphi '(a)$ and derivatives of $\varphi $ at points on $\partial U$ that are fixed by some iterate of $\varphi $. Our work depends upon a formula we establish for the essential spectral radius of any composition operator on the Hardy space $H^p$.

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Additional Information

Paul S. Bourdon
Affiliation: Department of Mathematics, Washington and Lee University, Lexington, Virginia 24450

Joel H. Shapiro
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

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

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