Resumen de Composition operators on the algebra of Dirichlet series

Manuel Domingo Contreras Márquez , Carlos Gómez Cabello, Luis Rodríguez Piazza

• The algebra of Dirichlet series A(C+) consists on those Dirichlet series convergent in the right half-plane 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+ giving rise to bounded composition operators C in A(C+) and denote this class by GA. We also characterise when the operator C is compact in A(C+). As a byproduct, we show that the weak compactness is equivalent to the compactness for C.

  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} in the class GA and strongly continuous semigroups of composition operators {Tt}, Tt f = f ◦ t , f ∈ A(C+). We conclude providing examples showing the differences between the symbols of bounded composition operators in A(C+) and the Hardy spaces of Dirichlet series Hp and H∞.