Volterra operators on the BMOA space of Dirichlet series

In this paper, we completely characterize the boundedness and compactness of Volterra operators on $BMO{A}_{\mathcal{D}}$, the BMOA space of Dirichlet series, and estimate their essential and weak essential norms. We further prove that compactness, weak compactness, and complete continuity are equivalent for these operators. In particular, every Dirichlet polynomial symbol induces a compact Volterra operator on $BMO{A}_{\mathcal{D}}$. We also prove that every such symbol induces a compact Volterra operator on the Bergman space of Dirichlet series ${\mathcal{A}}_{\mu }^1$ for every admissible measure $\mu$, extending the corresponding compactness theorem, previously established for $1<p<\mathrm{\infty }$, to the endpoint $p=1$.