Schatten class composition operators on the Hardy space of Dirichlet series and a comparison-type principle

We give necessary and sufficient conditions for a composition operator with Dirichlet series symbol to belong to the Schatten classes $S_p$ of the Hardy space ${\mathcal{H}}^2$ of Dirichlet series. For $p\ge 2$, these conditions lead to a characterization for the subclass of symbols with bounded imaginary parts. Finally, we establish a comparison-type principle for composition operators. Applying our techniques in conjunction with classical geometric function theory methods, we prove the analogue of the polygonal compactness theorem for ${\mathcal{H}}^2$ and we give examples of bounded composition operators with Dirichlet series symbols on ${\mathcal{H}}^p,p>0$.