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 $C_{\Phi }$ in $\mathcal{A}({\mathcal{C}}_{+})$ and denote this class by ${\mathcal{G}}_{\mathcal{A}}$. We also characterise when the operator $C_{\Phi }$ is compact in $\mathcal{A}(C_{+})$. 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 $\{T_t\}$, $T_{t}f=f\circ {\Phi }_t$, $f\in \mathcal{A}({\mathcal{C}}_{+})$. 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 ${\mathcal{H}}^{\infty }$.