Composition operators on the algebra of Dirichlet series

The algebra of Dirichlet series A(C₊) A ( C + ) consists on those Dirichlet series convergent in the right half-plane 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 Φ :C₊→ C₊ Φ : C + → C + giving rise to bounded composition operators CΦ C Φ in A(C₊) A ( C + ) and denote this class by GA G A . We also characterise when the operator CΦ C Φ is compact in A(C₊) A ( C + ) . As a byproduct, we show that the weak compactness is equivalent to the compactness for CΦ 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 \Φ ₜ\ { Φ t } in the class GA G A and strongly continuous semigroups of composition operators ₜ\ { T t } , Tₜf=f∘ Φ ₜ T t f = f ∘ Φ t , f∈ A(C₊) f ∈ A ( C + ) . We conclude providing examples showing the differences between the symbols of bounded composition operators in A(C₊) A ( C + ) and the Hardy spaces of Dirichlet series Hᵖ H p and H∞ H ∞ .