Mean growth of Koenigs eigenfunctions

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 $H^p$.