Estimate for Norm of a Composition Operator on the Hardy–Dirichlet Space

  • Perumal Muthukumar
  • Saminathan Ponnusamy
  • Hervé Queffélec
Article
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Abstract

By using the Schur test, we give some upper and lower estimates on the norm of a composition operator on \(\mathcal {H}^2\), the space of Dirichlet series with square summable coefficients, for the inducing symbol \(\varphi (s)=c_1+c_{q}q^{-s}\) where \(q\ge 2\) is a fixed integer. We also give an estimate on the approximation numbers of such an operator.

Keywords

Composition operator Hardy space Dirichlet series Schur test Zeta function 

Mathematics Subject Classification

Primary: 47B33 47B38 Secondary: 11M36 37C30 

Notes

Acknowledgements

The authors thank the referee for many useful comments. The first author thanks the Council of Scientific and Industrial Research (CSIR), India, for providing financial support in the form of a SPM Fellowship to carry out this research. The third author thanks the Indian Statistical Institute of Chennai for providing good and friendly working conditions in December 2015, when this collaboration was initiated. The second author is currently at ISI Chennai.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Stat-Math UnitIndian Statistical Institute (ISI), Chennai CentreChennaiIndia
  2. 2.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia
  3. 3.F-59 000 Lille, USTL, Laboratoire Paul Painlevé U.M.R. CNRS 8524Univ Lille Nord de FranceVilleneuve D’ascq CedexFrance

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