Hardy space of translated Dirichlet series

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