Hardy spaces of vector-valued Dirichlet series

Given a Banach space $X$ and $1\le p\le \mathrm{\infty }$, it is well known that the two Hardy spaces $H_p(\mathbb{T},X)$ ($\mathbb{T}$ the torus) and $H_p(\mathbb{D},X)$ ($\mathbb{D}$ the disk) have to be distinguished carefully. This motivates us to define and study two different types of Hardy spaces ${\mathcal{H}}_p(X)$ and ${\mathcal{H}}_p^{+}(X)$ of Dirichlet series $\sum_n a_n n^{-s}$withcoefficientsin$X$. We characterize them in terms of summing operators as well as holomorphic functions in infinitely many variables, and prove that they coincide whenever $X$ has the analytic Radon-Nikod'{y}m Property. Consequences are, among others, a vector-valued version of the Brother’s Riesz Theorem in the infinite-dimensional torus, and an answer to the question when ${\mathcal{H}}_1(X^{\ast })$ is a dual space.