Hardy space of translated Dirichlet series

Abstract

We study the Hardy space of translated Dirichlet series $\mathcal{H}_{+}$. It consists on those Dirichlet series $\sum a_n n^{-s}$ such that for some (equivalently, every) $1 \leq p < \infty$, the translation $\sum{a_{n}}n^{-(s+\frac{1}{\sigma})}$ belongs to the Hardy space $\mathcal{H}^{p}$ for every $\sigma>0$. We prove that this set, endowed with the topology induced by the seminorms $\left\{\Vert\cdot\Vert_{2,k}\right\}_{k\in\mathbb{N}}$ (where $\Vert\sum{a_{n}n^{-s}}\Vert_{2,k}$ is defined as $\big\Vert\sum{a_n n^{-(s+\frac{1}{k})}} \big\Vert_{\mathcal{H}^{2}}$), is a Fr\'echet space which is Schwartz and non nuclear. Moreover, the Dirichlet monomials $\{n^{-s}\}_{n \in \mathbb N}$ are an unconditional Schauder basis of $\mathcal H_+$. In the spirit of Gordon and Hedenmalm's work, we completely characterize the composition operator on the Hardy space of translated Dirichlet series. Moreover, we study the superposition operators on $\mathcal{H}_{+}$ and show that every polynomial defines an operator of this kind. We present certain sufficient conditions on the coefficients of an entire function to define a superposition operator. Relying on number theory techniques we exhibit some examples which do not provide superposition operators. We finally look at the action of the differentiation and integration operators on these spaces.