Composition Operators on Bohr-Bergman Spaces of Dirichlet Series

Abstract

For $\alpha \in \mathbb{R}$, let ${\mathcal{D}}_{\alpha }$ 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 ${\mathcal{H}}^2={\mathcal{D}}_0$ is extended to the Bergman case $\alpha >0$. These composition operators are generated by functions of the form $\Phi (s)=c_{0}s+\phi (s)$, where $c_0$ is a nonnegative integer and $\phi (s)$ 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 ${\mathcal{D}}_{\alpha }$ is mapped by the composition operator into a smaller space in the same scale. When $\alpha >1$, the space ${\mathcal{D}}_{\alpha }$ is mapped into a larger space in the same scale. Moreover, a partial description of the composition operators on the Dirichlet--Bergman spaces ${\mathcal{A}}^p$ for $1\le p<\infty$ are obtained, in addition to new partial results for composition operators on the Dirichlet--Hardy spaces ${\mathcal{H}}^p$ when $p$ is an odd integer.

Minor changes. Section 2 shortened

References in corpus (1)

• Multipliers on Hilbert Spaces of Dirichlet Series

Cited by in corpus (5)

• Composition operators on weighted Hilbert spaces of Dirichlet series
• Composition operators and embedding theorems for some function spaces of Dirichlet series
• Weak product spaces of Dirichlet series
• Topological structure of the space of composition operators on the Hardy space of Dirichlet series
• Volterra operators and Hankel forms on Bergman spaces of Dirichlet series