Integration Operators on Spaces of Dirichlet Series

We first study the Volterra operator V acting on spaces of Dirichlet series. We prove that V is bounded on the Hardy space \({\cal H}_0^p\) for any 0 < p ≤ ∞, and is compact on \({\cal H}_0^p\) for 1 <p ≤ ∞. Furthermore, we show that V is cyclic but not supercyclic on \({\cal H}_0^p\) for any 0 <p< ∞. Corresponding results are also given for V acting on Bergman spaces \({\cal H}_{w,0}^p\). We then study the Volterra type integration operators T_g. We prove that if T_g is bounded on the Hardy space \({{\cal H}^p}\), then it is bounded on the Bergman space \({\cal H}_w^p\).