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Composition operators and embedding theorems for some function spaces of Dirichlet series

  • Frédéric Bayart
  • Ole Fredrik Brevig
Article

Abstract

We observe that local embedding problems for certain Hardy and Bergman spaces of Dirichlet series are equivalent to boundedness of a class of composition operators. Following this, we perform a careful study of such composition operators generated by polynomial symbols \(\varphi \) on a scale of Bergman-type Hilbert spaces \(\mathscr {D}_\alpha \). We investigate the optimal \(\beta \) such that the composition operator \(\mathscr {C}_\varphi \) maps \(\mathscr {D}_\alpha \) boundedly into \(\mathscr {D}_\beta \). We also prove a new embedding theorem for the non-Hilbertian Hardy space \(\mathscr {H}^p\) into a Bergman space in the half-plane and use it to consider composition operators generated by polynomial symbols on \(\mathscr {H}^p\), finding the first non-trivial results of this type. The embedding also yields a new result for the functional associated to the multiplicative Hilbert matrix.

Mathematics Subject Classification

Primary 47B33 Secondary 30B50 30H10 30H20 

Notes

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Clermont Université, Université Blaise Pascal, Laboratoire de Mathématiques, UMR 6620-CNRSAubière cedexFrance
  2. 2.Department of Mathematical SciencesNorwegian University of Science and Technology (NTNU)TrondheimNorway

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