Composition Operators on Bohr-Bergman Spaces of Dirichlet Series

TL;DR

This paper extends the theory of composition operators on Dirichlet series spaces, revealing new phenomena in the mapping properties between different scales of these spaces, and provides partial characterizations for various related spaces.

Contribution

It generalizes the Gordon--Hedenmalm theorem to Dirichlet-Bergman spaces and uncovers new behaviors of composition operators depending on the parameter lpha.

Findings

For 0 < lpha < 1, composition operators map lpha spaces into smaller spaces.

For lpha > 1, they map into larger spaces within the same scale.

Partial descriptions of composition operators on ^p and ^p spaces are obtained.

Abstract

For $\alpha \in \mathbb{R}$, let $\mathscr{D}_\alpha$ denote the scale of Hilbert spaces consisting of Dirichlet series $f(s) = \sum_{n=1}^\infty a_n n^{-s}$ that satisfy $\sum_{n=1}^\infty |a_n|^2/[d(n)]^\alpha < \infty$. The Gordon--Hedenmalm Theorem on composition operators for $\mathscr{H}^2=\mathscr{D}_0$ is extended to the Bergman case $\alpha>0$. These composition operators are generated by functions of the form $\Phi(s) = c_0 s + \varphi(s)$, where $c_0$ is a nonnegative integer and $\varphi(s)$ is a Dirichlet series with certain convergence and mapping properties. For the operators with $c_0=0$ a new phenomenon is discovered: If $0 < \alpha < 1$, the space $\mathscr{D}_\alpha$ is mapped by the composition operator into a smaller space in the same scale. When $\alpha > 1$, the space $\mathscr{D}_\alpha$ is mapped into a larger space in the same scale. Moreover, a partial description of the composition operators on the Dirichlet--Bergman spaces $\mathscr{A}^p$ for $1 \leq p < \infty$ are obtained, in addition to new partial results for composition operators on the Dirichlet--Hardy spaces $\mathscr{H}^p$ when $p$ is an odd integer.