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Complete frequencies for Koenigs domains

Filippo Bracci, Eva A. Gallardo-Gutiérrez, Dmitry Yakubovich

TL;DR

This work advances the spectral analysis of semigroups of holomorphic self-maps of the disk by linking the completeness of exponential frequencies to the geometry and boundary dynamics of Koenigs domains. It provides a complete characterization of when $\mathcal{E}_\infty(\Omega)$ is weak-star dense in $H^\infty(\Omega)$ for non-elliptic semigroups, expressed via Carathéodory prime ends, Cantor combs, and maximal contact arcs. It also develops robust density results in $H^p(\Omega)$, with key cases for half-planes, strips, and logarithmic starlike-at-infinity domains, and gives necessary conditions for $p$-completeness, including explicit counterexamples. The methods combine analytic function theory, Laplace-transform techniques, and Koenigs-domain geometry to connect approximation by exponentials with the dynamical properties of the semigroup. Overall, the paper deepens understanding of when exponential systems provide complete spectral representations in Hardy spaces tied to semigroup dynamics, with potential implications for spectral synthesis and operator theory in complex analysis.

Abstract

We provide a complete characterization of those non-elliptic semigroups of holomorphic self-maps of the unit disc for which the linear span of eigenvectors of the generator of the corresponding semigroup of composition operators is weak-star dense in $H^\infty$. We also give some necessary and some sufficient conditions for completeness in $H^p$. This problem is equivalent to the completeness of the corresponding exponential functions in $H^\infty$ (in the weak-star sense) or in $H^p$ of the Koenigs domain of the semigroup. As a tool needed for the results, we introduce and study discontinuities of semigroups of holomorphic self-maps of the unit disc.

Complete frequencies for Koenigs domains

TL;DR

This work advances the spectral analysis of semigroups of holomorphic self-maps of the disk by linking the completeness of exponential frequencies to the geometry and boundary dynamics of Koenigs domains. It provides a complete characterization of when is weak-star dense in for non-elliptic semigroups, expressed via Carathéodory prime ends, Cantor combs, and maximal contact arcs. It also develops robust density results in , with key cases for half-planes, strips, and logarithmic starlike-at-infinity domains, and gives necessary conditions for -completeness, including explicit counterexamples. The methods combine analytic function theory, Laplace-transform techniques, and Koenigs-domain geometry to connect approximation by exponentials with the dynamical properties of the semigroup. Overall, the paper deepens understanding of when exponential systems provide complete spectral representations in Hardy spaces tied to semigroup dynamics, with potential implications for spectral synthesis and operator theory in complex analysis.

Abstract

We provide a complete characterization of those non-elliptic semigroups of holomorphic self-maps of the unit disc for which the linear span of eigenvectors of the generator of the corresponding semigroup of composition operators is weak-star dense in . We also give some necessary and some sufficient conditions for completeness in . This problem is equivalent to the completeness of the corresponding exponential functions in (in the weak-star sense) or in of the Koenigs domain of the semigroup. As a tool needed for the results, we introduce and study discontinuities of semigroups of holomorphic self-maps of the unit disc.
Paper Structure (25 sections, 49 theorems, 119 equations)

This paper contains 25 sections, 49 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Semigroups in the unit disc, Koenigs functions and Koenigs domains
  4. Hardy spaces and weak-star topology
  5. Infinitesimal generators of composition operators semigroups
  6. Frequencies of semigroups in Hardy spaces
  7. Approximation by exponentials in some starlike at infinity domains
  8. The half-plane and the strip
  9. Logarithmic starlike at infinity domains
  10. Weak-star density for Koenigs domains
  11. Characterization of Koenigs domains $\Omega$ such that $\hbox{Int}(\overline{\Omega})=\Omega$
  12. Prime ends and impressions
  13. Maximal contact arcs
  14. Discontinuities of Koenigs functions
  15. Characterization
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $(\varphi_t)_{t\geq 0}$ be a hyperbolic semigroup in $\mathbb D$ with Denjoy-Wolff point $\tau\in\partial\mathbb D$. Let $h$ be the Koenigs function, $\Omega=h(\mathbb D)$ the Koenigs domain and let $\psi:I\to \mathbb R$ be the defining function of $\Omega$. The following conditions are equivale Moreover, if one---and hence any---of the above conditions (1)--(4) holds, then $(\varphi_t)_{t\geq

Theorems & Definitions (106)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • ...and 96 more