Composition operators on the algebra of Dirichlet series

The algebra of Dirichlet series \(\mathcal {A}({{\mathbb {C}}}_+)\) consists on those Dirichlet series convergent in the right half-plane \({{\mathbb {C}}}_+\) and which are also uniformly continuous there. This algebra was recently introduced by Aron, Bayart, Gauthier, Maestre, and Nestoridis. We describe the symbols \(\Phi :{{\mathbb {C}}}_+\rightarrow {{\mathbb {C}}}_+\) giving rise to bounded composition operators \(C_{\Phi }\) in \(\mathcal {A}({{\mathbb {C}}}_+)\) and denote this class by \(\mathcal {G}_{\mathcal {A}}\). We also characterise when the operator \(C_{\Phi }\) is compact in \(\mathcal {A}({{\mathbb {C}}}_+)\). As a byproduct, we show that the weak compactness is equivalent to the compactness for \(C_{\Phi }\). Next, the closure under the local uniform convergence of several classes of symbols of composition operators in Banach spaces of Dirichlet series is discussed. We also establish a one-to-one correspondence between continuous semigroups of analytic functions \(\{\Phi _t\}\) in the class \(\mathcal {G}_{\mathcal {A}}\) and strongly continuous semigroups of composition operators \(\{T_t\}\), \(T_tf=f\circ \Phi _t\), \(f\in \mathcal {A}({{\mathbb {C}}}_+)\). We conclude providing examples showing the differences between the symbols of bounded composition operators in \(\mathcal {A}({{\mathbb {C}}}_+)\) and the Hardy spaces of Dirichlet series \(\mathcal {H}^p\) and \(\mathcal {H}^{\infty }\).