Spaces of Dirichlet series with the complete Pick property

TL;DR

This paper characterizes when certain Dirichlet series spaces are complete Pick spaces, explores their isomorphisms with Drury-Arveson spaces, and analyzes their multiplier algebras, revealing deep structural connections.

Contribution

It provides a complete characterization of Dirichlet series spaces with the complete Pick property and establishes their weak isomorphisms with Drury-Arveson spaces, linking their multiplier algebras.

Findings

Many Dirichlet series spaces are weakly isomorphic to Drury-Arveson spaces.

Multiplier algebras of these Dirichlet spaces are unitarily equivalent to those of $H^2_d$.

Every complete Pick algebra is a quotient of the multiplier algebra of these Dirichlet spaces.

Abstract

We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form $k(s,u) = \sum a_n n^{-s-\bar u}$, and characterize when such a space is a complete Pick space. We then discuss what it means for two reproducing kernel Hilbert spaces to be "the same", and introduce a notion of weak isomorphism. Many of the spaces we consider turn out to be weakly isomorphic as reproducing kernel Hilbert spaces to the Drury-Arveson space $H^2_d$ in $d$ variables, where $d$ can be any number in $\{1,2,\ldots, \infty\}$, and in particular their multiplier algebras are unitarily equivalent to the multiplier algebra of $H^2_d$. Thus, a family of multiplier algebras of Dirichlet series are exhibited with the property that every complete Pick algebra is a quotient of each member of this family. Finally, we determine precisely when such a space of Dirichlet series is weakly isomorphic to $H^2_d$ and when its multiplier algebra is isometrically isomorphic to $Mult(H^2_d)$.