Volterra operator acting on Bergman spaces of Dirichlet series

Since their introduction in 1997, the Hardy spaces of Dirichlet series have been broadedly and deeply studied. The increasing interest sparked by these Banach spaces of Dirichlet series motivated the introduction of new such spaces, as the Bergman spaces of Dirichlet series Aᵖμ here considered, where μ is a probability measure on (0,∞). Similarly, recent lines of research have focused their attention in the study of some classical operators acting on these spaces, as it is the case of the Volterra operator Tg. In this work, we introduce a new family of Bloch spaces of Dirichlet series, the Blochμ-spaces, and study some of its most essential properties. Using these spaces we are able to provide a sufficient condition for the Volterra operator Tg to act boundedly on the Bergman spaces Aᵖμ. We also establish a necessary condition for a specific choice of the probability measures μ. Sufficient and necessary conditions for compactness are also proven. The membership to Schatten classes is studied as well. Eventually, a radicality result is established for Bloch spaces in the polydisc.