Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Concrete Operators

Ed. by Ross, William / Mashreghi, Javad

1 Issue per year

Mathematical Citation Quotient (MCQ) 2017: 0.34

Open Access
See all formats and pricing
More options …

Iteration of Composition Operators on small Bergman spaces of Dirichlet series

Jing Zhao
Published Online: 2018-05-24 | DOI: https://doi.org/10.1515/conop-2018-0003


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


  • [1] M. Bailleul, O. F. Brevig, The composition operators on Bohr-Bergman spaces of Dirichlet series, Ann. Acad. Sci. Fenn. Math. 1, 41 (2016)Google Scholar

  • [2] J. Gordon, H. Hedenmalm, The composition operators on the space of Dirichlet series with square summable coefficients, Michigan Math. J. 2, 46 (1999)Google Scholar

  • [3] H. Hedenmalm, P. Lindqvist, K. Seip, A Hilbert space of Dirichlet series and systems of dilated functions in L2(0, 1), Duke Math. J. 1, 86 (1997)Google Scholar

  • [4] J. L. Nicolas, Sur la distribution des nombres entiers ayant une quantité fixée de facteurs premiers, Acta Arith. 3, 44 (1984)Google Scholar

  • [5] J. F. Olsen, Local properties of Hilbert spaces of Dirichlet series, J. Funct. Anal. 9, 261 (2011)Web of ScienceGoogle Scholar

  • [6] H. Quffélec, K. Seip, Approximation numbers of composition operators on the M2 space of Dirichlet series, J. Funct. Anal. 6, 268 (2015)Web of ScienceGoogle Scholar

  • [7] G. Tenenbaum, Introduction to analytic and probabilistic number theory, (Cambridge University Press, Cambridge, 1995)Google Scholar

  • [8] P. Turán, On a theorem of Hardy and Ramanujan, J. London Math. Soc. 4, 274 (1934)Google Scholar

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.

Export Citation

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

Comments (0)

Please log in or register to comment.
Log in