Semigroups of composition operators on Hardy spaces of Dirichlet series

TL;DR

This paper establishes a correspondence between continuous semigroups of analytic functions in the Gordon-Hedenmalm class and strongly continuous semigroups of composition operators on Hardy spaces of Dirichlet series, extending results across different p-norms.

Contribution

It characterizes the infinitesimal generators of these semigroups and describes their dynamical properties, providing a comprehensive framework for understanding composition operators in this setting.

Findings

One-to-one correspondence between semigroups of functions and composition operators.

No non-trivial strongly continuous semigroups in ty-Hardy space.

Infinitesimal generators characterized as Dirichlet series mapping into their closure.

Abstract

We consider continuous semigroups of analytic functions $\{\Phi_t\}_{t\geq0}$ in the so-called Gordon-Hedenmalm class $\mathcal{G}$, that is, the family of analytic functions $\Phi:\mathbb C_+\to \mathbb C_+$ giving rise to bounded composition operators in the Hardy space of Dirichlet series $\mathcal{H}^2$. We show that there is a one-to-one correspondence between continuous semigroups $\{\Phi_{t}\}_{t\geq0}$ in the class $\mathcal G$ and strongly continuous semigroups of composition operators $\{T_t\}_{t\geq0}$, where $T_t(f)=f\circ\Phi_t$, $f\in\mathcal{H}^2$. We extend these results for the range $p\in[1,\infty)$. For the case $p=\infty$, we prove that there is no non-trivial strongly continuous semigroup of composition operators in $\mathcal{H}^\infty$. We characterize the infinitesimal generators of continuous semigroups in the class $\mathcal G$ as those Dirichlet series sending $\mathbb C_{+}$ into its closure. Some dynamical properties of the semigroups are obtained from a description of the Koenigs map of the semigroup.