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

TL;DR

This paper investigates the norms of composition operators on the Hardy space of Dirichlet series, establishing subordination principles, characterizing maximal norm operators, and improving bounds for specific symbols.

Contribution

It introduces two subordination principles for these operators, generalizes Shapiro's result, and refines norm bounds for particular Dirichlet series symbols.

Findings

Two subordination principles characterize maximal norm composition operators.

Generalization of Shapiro's classical Hardy space result.

Improved upper and lower norm bounds for specific symbols.

Abstract

We consider composition operators $\mathscr{C}_\varphi$ on the Hardy space of Dirichlet series $\mathscr{H}^2$, generated by Dirichlet series symbols $\varphi$. 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 $\varphi$. The other concerns general symbols, and is based on a geometrical condition on the boundary values of $\varphi$. 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 $\varphi(s) = c + r2^{-s}$. A number of other examples are given.