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Concrete Operators

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Mathematical Citation Quotient (MCQ) 2016: 0.38


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Iteration of Composition Operators on small Bergman spaces of Dirichlet series

Jing Zhao
  • Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
  • Other articles by this author:
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Published Online: 2018-05-24 | DOI: https://doi.org/10.1515/conop-2018-0003

Abstract

The Hilbert Spaces H<sub>w</sub> Consisiting Of Dirichlet Series F(S) = Σ<sup>∞</sup> <sub>n=1 </sub>A<sub>n</sub>n<sup>−S</sup> that satisfty Σ<sup>∞</sup> <sub>n=1 | a<sub>n</sub>|<sup>2</sup>/w<sub>n</sub> < ∞ with {W<sub>n</sub>}<sub>n</sub> of Average Order Log<sub>j</sub> 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 H<sub>w</sub> From an iterative point of view. By that theorem, the composition operators are generated by functions of the form Ф(S) = C<sub>0</sub>S + ϕ (S), Where C<sub>0</sub> is a nonnegative integer and ϕ is a Dirichlet series with certain convergence and mapping properties. The iterative phenomenon takes place when C<sub>0</sub> = 0. It is verified for every integer j ≥ 1, real α > 0 and {W<sub>n</sub>}<sub>n</sub> having Average Order (Log<sup>+</sup><sub>J</sub> n)<sup>α</sup>, that the composition operators map H<sub>w</sub> into a scale of H<sub>w</sub> with W<sup>'</sup><sub>n</sub> having average order (Log<sup>+</sup><sup>J +1 </sup>n)<sup>α </sup>. 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

MSC 2010: 47B33; 11N37

References

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About the article

Received: 2017-05-21

Accepted: 2017-10-23

Published Online: 2018-05-24


Citation Information: Concrete Operators, Volume 5, Issue 1, Pages 24–34, ISSN (Online) 2299-3282, DOI: https://doi.org/10.1515/conop-2018-0003.

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© 2018 Jing Zhao, published by Sciendo. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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