Composition operators on the algebra of Dirichlet series

Abstract: The algebra of Dirichlet series $\mathcal{A}(\mathcal{C}{+})$ consists on those Dirichlet series convergent in the right half-plane $\mathcal{C}{+}$ and which are also uniformly continuous there. This algebra was recently introduced by Aron, Bayart, Gauthier, Maestre, and Nestoridis. We describe the symbols $\Phi:\mathcal{C}{+}\to\mathcal{C}{+}$ giving rise to bounded composition operators $\mathit{C}{\Phi}$ in $\mathcal{A}(\mathcal{C}{+})$ and denote this class by $\mathcal{G}{\mathcal{A}}$. We also characterise when the operator $\mathit{C}{\Phi}$ is compact in $\mathcal{A}(\mathit{C}{+})$. As a byproduct, we show that the weak compactness is equivalent to the compactness for $\mathit{C}{\Phi}$. 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 ${T{t}}$, $T_{t}f=f\circ\Phi_{t}$, $f\in\mathcal{A}(\mathcal{C}{+})$. We conclude providing examples showing the differences between the symbols of bounded composition operators in $\mathcal{A}(\mathcal{C}{+})$ and the Hardy spaces of Dirichlet series $\mathcal{H}^{p}$ and $\mathcal{H}^{\infty}$.