Composition operators on Bohr-Bergman spaces of Dirichlet series

Abstract

Abstract. For α ∈ R, let Dα denote the scale of Hilbert spaces consisting of Dirichlet series f (s) =The Gordon-Hedenmalm Theorem on composition operators for H 2 = D 0 is extended to the Bergman case α > 0. These composition operators are generated by functions of the form Φ(s) = c 0 s+ϕ(s), where c 0 is a nonnegative integer and ϕ(s) is a Dirichlet series with certain convergence and mapping properties. For the operators with c 0 = 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 DirichletHardy spaces H p when p is an odd integer.

Weak Product Spaces of Dirichlet Series

“…In fact, suppose that n = j p It should also be pointed out that decomposing Dirichlet series (or power series on the infinite polydisc) into homogeneous subseries is not a new idea. It dates back at least to Bohnenblust-Hille [6], and has recently been applied to obtain results for composition operators on spaces of Dirichlet series [3] as well as L 1 -estimates for Dirichlet polynomials [8].…”

Iteration of Composition Operators on small Bergman spaces of Dirichlet series

“…Notice that d α+1 (n) has average order (log n) α [11] and d (n) α has average order (log n) 2 α −1 . 2…”

“…As in [2], we assume F •Φ(s) = ∞ n=2 b n n −s . To avoid negative arguments in the j -fold logarithm, we shall equip χ ∈ T ∞ with an indicator function with respect to the value of Ω(n) by defining…”

“…The author is supported by the Research Council of Norway grant 227768. 1 We observe from the proof of this extension (see [2,Theorem 1]) that D α are actually mapped into weighted Hilbert spaces that consist of Dirichlet series F (s) = ∞ n=1 a n n −s satisfying…”

“…The set of such Φ is called the Gordon-Hedenmalm class and denoted by G . With the same convergence and mapping properties, the Gordon-Hedenmalm theorem was extended to the weighted Hilbert spaces D α which consists of Dirichlet series F (s) = ∞ n=1 a n n −s that satisfy ∞ n=1 |a n | 2 /d (n) α < ∞ in [2]. Here d (n) is the divisor function which counts the number of divisors of n and α > 0.…”