Spaces of Dirichlet series with the complete Pick property
John E. McCarthy, Orr Shalit
TL;DR
This work develops a framework to understand complete Pick spaces among reproducing kernel Hilbert spaces, with a focus on spaces of Dirichlet series. It leverages the McCullough–Quiggin theorem and the concept of weak isomorphism to compare Dirichlet-series RKHS with the Drury–Arveson spaces, culminating in a universal Dirichlet-space construction that is weakly isomorphic to $H^2_\infty$ and yields unitary equivalence of multiplier algebras. The paper then identifies precise conditions under which such Dirichlet spaces are universal, namely when the logarithms of the indexing integers are $\mathbb Q$-linearly independent, and proves non-universality in the dependent case. Overall, it provides a universal representation for complete Pick algebras via Dirichlet-series kernels and clarifies the structure of their multiplier algebras, with connections to prime-based embeddings and Drury–Arveson space theory.
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 $Mult(H^2_d)$.