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BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access May 24, 2018

Iteration of Composition Operators on small Bergman spaces of Dirichlet series

  • Jing Zhao EMAIL logo
From the journal Concrete Operators


The Hilbert spaces ℋw consisiting of Dirichlet series F(s)=n=1ann-s that satisfty 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 verified for every integer j ⩾ 1, real α > 0 and {wn}n having average order (logj+n)α, that the composition operators map ℋw into a scale of ℋw with wn having average order (logj+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.

MSC 2010: 47B33; 11N37


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Received: 2017-05-21
Accepted: 2017-10-23
Published Online: 2018-05-24

© 2018 Jing Zhao, published by De Gruyter

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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