Multipliers for Hardy spaces of Dirichlet series

TL;DR

This paper characterizes the multipliers between Hardy spaces of Dirichlet series, analyzing their properties and spectra, and connects these results to spaces of holomorphic functions in infinitely many variables.

Contribution

It provides a complete characterization of multipliers between Hardy spaces of Dirichlet series for all p,q, and studies their operator properties.

Findings

Characterization of multipliers between  and  Hardy spaces of Dirichlet series.

Analysis of the norm, essential norm, and spectrum of associated multiplication operators.

Extension of results to Hardy spaces of holomorphic functions in infinitely many variables.

Abstract

We characterize the space of multipliers from the Hardy space of Dirichlet series $\mathcal H_p$ into $\mathcal H_q$ for every $1 \leq p,q \leq \infty$. For a fixed Dirichlet series, we also investigate some structural properties of its associated multiplication operator. In particular, we study the norm, the essential norm, and the spectrum for an operator of this kind. We exploit the existing natural identification of spaces of Dirichlet series with spaces of holomorphic functions in infinitely many variables and apply several methods from complex and harmonic analysis to obtain our results. As a byproduct we get analogous statements on such Hardy spaces of holomorphic functions.