Composition operators on the algebra of Dirichlet series

Abstract

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