K\"onigs maps and commutants of composition operators on the Hardy-Hilbert space of Dirichlet series

TL;DR

This paper investigates the properties of composition operators on the Hardy-Hilbert space of Dirichlet series, focusing on K"onigs maps, spectral and dynamical characteristics, and conditions for minimal commutants.

Contribution

It introduces a method to find K"onigs maps for symbols of composition operators and explores their implications on spectral and commutant properties.

Findings

Established a procedure to find K"onigs maps for certain symbols.

Analyzed the spectrum and dynamical behavior of the composition operators.

Identified conditions under which these operators have minimal commutants.

Abstract

Let $\varphi$ be a holomorphic map which is a symbol of a bounded composition operator $C_\varphi$ acting on the Hardy-Hilbert space of Dirichlet series. We find a K\"onigs map for $\varphi$. We then deduce several applications on $C_\varphi$ (e.g. on its spectrum, on its dynamical properties). In particular, we study for a large class of symbols $\varphi$ if the associated composition operator has a minimal commutant.