Composition operators on the algebra of Dirichlet series

Abstract 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: { {C}}_+ { {C}}_+ Φ: C + → C + giving rise to bounded composition operators C C Φ in A ({ {C}}_+) A (C +) and denote this class by G ₀ G A. We also characterise when the operator C C Φ is compact in A ({ {C}}_+) A (C +). As a byproduct, we show that the weak compactness is equivalent to the compactness for C 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 \ ₜ\ Φ t in the class G ₀ G A and strongly continuous semigroups of composition operators \Tₜ\ T t, Tₜf=f 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 ∞.