On frequencies of parabolic Koenigs domains

TL;DR

This paper characterizes the point spectrum of the generator of composition semigroups on Hardy spaces induced by parabolic semigroups, linking spectral properties to geometric features of the Koenigs domain.

Contribution

It provides a complete description of the point spectrum for the generator of composition operators associated with parabolic semigroups, extending previous results to broader geometric settings.

Findings

Complete spectral characterization when the Koenigs domain contains an angular sector.

Necessary and sufficient conditions for frequencies based on geometric properties of the domain.

Extension of Betsakos's work to more general Koenigs domains.

Abstract

Let $(\varphi_t)_{t\geq 0}$ 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 $(C_{\varphi_t})_{t\geq 0}$ 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 first 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 $\Omega$. Furthermore, assuming that $\Omega$ is convex, we also obtain necessary conditions for $\lambda$ to be a frequency of $\Omega$. Using these, we are able to completely describe $\sigma_p(\Delta\mid_{H^p})$ in a broad range of situations e.g. when $\Omega$ contains an angular sector. We conclude with some consequences regarding the spectrum of the composition operators $(C_{\varphi_t})_{t\geq 0}$. These results extend a previous work of Betsakos on hyperbolic semigroups.