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Complex Symmetry of Composition Operators on Hilbert Spaces of Entire Dirichlet Series

  • Minh Luan Doan
  • Camille Mau
  • Le Hai KhoiEmail author
Article
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Abstract

A criterion for boundedness of composition operators acting on a class of Hilbert spaces of entire Dirichlet series, namely the class \(\mathcal {H}(E, \beta _{S})\), was obtained in Hou et al. (J. Math. Anal. Appl. 401: 416–429, 2013) for those spaces that do not contain non-zero constant functions, while other possibilities were not studied. In this paper, we first provide a complete characterization of boundedness of composition operators on any space \(\mathcal {H}(E, \beta _{S})\), which may or may not contain constant functions. We then study complex symmetry of composition operators on \(\mathcal {H}(E, \beta _{S})\), via analysis of composition conjugations.

Keywords

Hilbert space Entire Dirichlet series Composition operator Conjugation Complex symmetry 

Mathematics Subject Classification (2010)

30E20 30D50 

Notes

Funding Information

The second-named author was supported in part by the CN Yang Scholars Programme, Nanyang Technological University. The third-named author was supported in part by MOE’s AcRF Tier 1 grant M4011724.110 (RG128/16).

References

  1. 1.
    Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bourdon, P.S., Waleed Noor, S.: Complex symmetry of invertible composition operators. J. Math. Anal. Appl. 429, 105–110 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chevrot, N., Fricain, E., Timotin, D.: The characteristic function of a complex symmetric contraction. Proc. Am. Math. Soc. 135, 2877–2886 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cowen, C.C., MacCluer, B.D.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995)zbMATHGoogle Scholar
  5. 5.
    Garcia, S.R., Putinar, M.: Complex symmetric operators and applications. Trans. Am. Math. Soc. 358, 1285–1315 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Garcia, S., Wogen, W.: Some new classes of complex symmetric operators. Trans. Am. Math. Soc. 362, 6065–6077 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hardy, G.H., Riesz, M.: The General Theory of Dirichlet’s Series. Stechert-hafner, Inc., New York (1964)zbMATHGoogle Scholar
  8. 8.
    Hou, X., Hu, B., Khoi, L.H.: Hilbert spaces of entire Dirichlet series and composition operators. J. Math. Anal. Appl. 401, 416–429 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hu, B., Khoi, L.H.: Numerical range of composition operators on Hilbert spaces of entire Dirichlet series. In: Son, L. H., Tutscheke, W. (eds.) Interactions between Real and Complex Analysis, pp 285–299. Science and Technology Publication House, Hanoi (2012)Google Scholar
  10. 10.
    Hu, B., Khoi, L.H., Zhao, R.: Topological structure of the spaces of composition operators on Hilbert spaces of Dirichlet series. Z. Anal. Anwend. 35, 267–284 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lim, R., Khoi, L.H.: Complex symmetric weighted composition operators on \(\mathcal {H}_{\gamma }(\mathbb {D})\). J. Math. Anal. Appl. 464, 101–118 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mandelbrojt, S.: Séries de Dirichlet. Principes Et Méthodes Monographies internationales de mathématiques modernes, vol. 11. Gauthier-Villars, Paris (1969)Google Scholar
  13. 13.
    Pólya, G.: On an integral function of an integral function. J. Lond. Math. Soc. s1–1, 12 (1926)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Reddy, A.R.: On entire Dirichlet series of zero order. Tôhoku Math. J. 18, 144–155 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ritt, J.F.: On certain points in the theory of Dirichlet series. Am. J. Math. 50, 73–86 (1928)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wang, M., Yao, X.: Some properties of composition operators on Hilbert spaces of Dirichlet series. Complex Var. Elliptic Equ. 60, 992–1004 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Notre DameNotre DameUSA
  2. 2.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological University (NTU)SingaporeSingapore

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