Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 129-142

COMPOSITION OPERATORS ON BOHR-BERGMAN SPACES OF DIRICHLET SERIES

Maxime Bailleul and Ole Fredrik Brevig

Univ Lille-Nord-de-France UArtois, Laboratoire de Mathématiques de Lens EA 2462
Féedération CNRS Nord-Pas-de-Calais FR 2956, F-62 300 Lens, France; maxime.bailleul 'at' euler.univ-artois.fr

Norwegian University of Science and Technology (NTNU), Department of Mathematical Sciences
NO-7491 Trondheim, Norway; ole.brevig 'at' math.ntnu.no

Abstract. For α ∈ R, let D_α denote the scale of Hilbert spaces consisting of Dirichlet series f(s) = ∑_n=1^∞ a_nn^-s that satisfy ∑_n=1^∞ |a_n|²/[d(n)]sup>α < ∞. The Gordon-Hedenmalm Theorem on composition operators for H² = D₀ is extended to the Bergman case α > 0. These composition operators are generated by functions of the form Φ(s) = c₀s + φ(s), where c₀ is a nonnegative integer and φ(s) is a Dirichlet series with certain convergence and mapping properties. For the operators with c₀ =0 a new phenomenon is discovered: If 0 < α < 1, the space D_α is mapped by the composition operator into a smaller space in the same scale. When α > 1, the space D_α is mapped into a larger space in the same scale. Moreover, a partial description of the composition operators on the Dirichlet-Bergman spaces A^p for 1 ≤ p < ∞ are obtained, in addition to new partial results for composition operators on the Dirichlet-Hardy spaces H^p when p is an odd integer.

2010 Mathematics Subject Classification: Primary 47B33; Secondary 30B50.

Key words: Composition operators, Dirichlet series, Bergman spaces.

Reference to this article: M. Bailleul and O.F. Brevig: Composition operators on Bohr-Bergman spaces of Dirichlet series. Ann. Acad. Sci. Fenn. Math. 41 (2016), 129-142.

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doi:10.5186/aasfm.2016.4104