Composition operators on Bohr-Bergman spaces of Dirichlet series

Abstract

For α ∈ R, let Dα denote the scale of Hilbert spaces consisting of Dirichlet series f(s) = P∞ n=1 ann −s that satisfy P∞ n=1 |an| 2/[d(n)]α < ∞. The Gordon–Hedenmalm Theorem on composition operators for H 2 = 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 Dα is mapped by the composition operator into a smaller space in the same scale. When α > 1, the space Dα 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.