Volterra operator acting on Bergman spaces of Dirichlet series

Abstract

Since their introduction in 1997, Hardy spaces of Dirichlet series have been broadly studied. The increasing interest which they sparked motivated the introduction of new such spaces, as the Bergman spaces $A_{\mu }^p$ considered here, with μ a probability measure. Similarly, recent lines of research have focused on the study of some classical operators acting on these spaces, like the Volterra operator $V_g$. In this work, we introduce a new family of spaces of Dirichlet series, the ${\text{Bloch}}_{\mu }$-spaces. We can provide, in terms of those spaces, a sufficient condition for this Volterra operator to act boundedly on the spaces $A_{\mu }^p$. We also establish a necessary condition for a specific choice of μ. Sufficient and necessary conditions for compactness are also proven. The non-membership in Schatten classes is established, as well as a radicality result for some Bloch space.