Semigroups of composition operators on Hardy spaces of Dirichlet series

Abstract

We consider continuous semigroups of analytic functions $\{{\Phi }_t{\}}_{t\ge 0}$ 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 $\{{\Phi }_t{\}}_{t\ge 0}$ in the class $\mathcal{G}$ and strongly continuous semigroups of composition operators $\{T_t{\}}_{t\ge 0}$, 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.

33 pages