Iteration of composition operators on small Bergman spaces of Dirichlet series

Abstract

The Hilbert spaces Hw consisiting of Dirichlet series F(s) = P∞ n=1 an n −s that satisfty P∞ n=1 |an| 2 /wn < ∞, with {wn}n of average order logj n (the j-fold logarithm of n), can be embedded into certain small Bergman spaces. Using this embedding, we study the Gordon–Hedenmalm theorem on such Hw from an iterative point of view. By that theorem, the composition operators are generated by functions of the form Φ(s) = c0s + ϕ(s), where c0 is a nonnegative integer and ϕ is a Dirichlet series with certain convergence and mapping properties. The iterative phenomenon takes place when c0 = 0. It is veri ed for every integer j > 1, real α > 0 and {wn}n having average order (log+ j n) α , that the composition operators map Hw into a scale of Hw′ with w ′ n having average order (log+ j+1 n) α . The case j = 1 can be deduced from the proof of the main theorem of a recent paper of Bailleul and Brevig, and we adopt the same method to study the general iterative step.