Norms of composition operators on the $H^2$ space of Dirichlet series

Abstract

We consider composition operators ${\mathcal{C}}_{\phi }$ on the Hardy space of Dirichlet series ${\mathcal{H}}^2$, generated by Dirichlet series symbols $\phi$. We prove two different subordination principles for such operators. One concerns affine symbols only, and is based on an arithmetical condition on the coefficients of $\phi$. The other concerns general symbols, and is based on a geometrical condition on the boundary values of $\phi$. Both principles are strict, in the sense that they characterize the composition operators of maximal norm generated by symbols having given mapping properties. In particular, we generalize a result of J. H. Shapiro on the norm of composition operators on the classical Hardy space of the unit disc. Based on our techniques, we also improve the recently established upper and lower norm bounds in the special case that $\phi (s)=c+r{2}^{-s}$. A number of other examples are given.

arXiv commentThis paper has been accepted for publication in Journal of Functional Analysis

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Cited by in corpus (5)

• A mean counting function for Dirichlet series and compact composition operators
• An extension of Bohr's theorem
• Schatten class composition operators on the Hardy space of Dirichlet series and a comparison-type principle
• Orthogonal decomposition of composition operators on the $H^2$ space of Dirichlet series
• Topological structure of the space of composition operators on the Hardy space of Dirichlet series