Composition operators on the algebra of Dirichlet series

• Manuel D. Contreras ^[1] ; Carlos Gómez-Cabello ^[1] ; Luis Rodríguez-Piazza ^[1]
  1. [1] Universidad de Sevilla

     Universidad de Sevilla

     Sevilla, España

     
• Localización: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas ( RACSAM ), ISSN-e 1578-7303, Vol. 118, Nº. 4, 2024
• Idioma: inglés
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• Resumen

  • 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∞.