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 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.

Keywords

• composition operators
• iteration
• bergman spaces
• dirichlet series
• 47b33
• 11n37