Multipliers for Hardy spaces of Dirichlet series

We characterise the space of multipliers from the Hardy space of Dirichlet series ℋ p into ℋ q for every 1≤p,q≤∞. For a fixed Dirichlet series, we also analyse some structural properties of its associated multiplication operator. In particular, we study the norm, the essential norm, and the spectrum for an operator of this kind. We exploit the existing natural identification of spaces of Dirichlet series with spaces of holomorphic functions in infinitely many variables and apply several methods from complex and harmonic analysis to obtain our results. As a byproduct we get analogous statements on such Hardy spaces of holomorphic functions.