On frequencies of parabolic Koenigs domains

Abstract

Let $({\phi }_t{)}_{t\ge 0}$ be a parabolic semigroup of analytic functions on $\mathbb{D}$, with Koenigs function $h$ and Koenigs domain $\Omega =h(\mathbb{D})$. We study the point spectrum ${\sigma }_p(\Delta {\mid }_{H^p})$ of $\Delta$, the infinitesimal generator of the $C_0$-semigroup $(C_{{\phi }_t}{)}_{t\ge 0}$ of composition operators on $H^p$. This reduces to characterizing the frequencies of $\Omega$. That is, those $\lambda \in \mathbb{C}$ such that $e^{\lambda h}\in H^p$. We derive containment relations for ${\sigma }_p(\Delta {\mid }_{H^p})$ and provide sufficient conditions for its complete characterization. Our approach relies heavily on the geometric properties of $\Omega$ and on careful estimates of the harmonic measure of some boundary subsets of $\Omega$. We conclude with some consequences regarding the spectrum of the composition operators $(C_{{\phi }_t}{)}_{t\ge 0}$. These results extend a previous work of Betsakos on hyperbolic semigroups.

Accepted for publication in Mathematische Zeitschrift, 32 pages