Iteration of Composition Operators on small Bergman spaces of Dirichlet series

Abstract The Hilbert spaces ℋw consisiting of Dirichlet series F(s)=∑n=1∞ann-s $F(s) = \sum\nolimits_{n = 1}^\infty {{a_n}{n^{ - s}}}$ that satisfty ∑n=1∞|an|2/wn<∞ ${\sum\nolimits_{n = 1}^\infty {\left| {{a_n}} \right|} ^2}/{w_n} < \infty $ 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 verified for every integer j ⩾ 1, real α > 0 and {wn}n having average order (logj+n)α ${(\log _j^ + n)^\alpha }$ , that the composition operators map ℋw into a scale of ℋw’ with w’n having average order (logj+1+n)α ${(\log _{j + 1}^ + n)^\alpha }$ . 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.