Title

Spaces of Dirichlet series with the complete Pick property

Authors

John E. McCarthy, Washington University in St LouisFollow
Orr Moshe Shalit

Author's School

Arts & Sciences

Author's Department

Mathematics

Document Type

Article

Publication Date

6-2017

Originally Published In

McCarthy, J.E. & Shalit, O.M. Isr. J. Math. (2017) 220(2): 509-530. doi:10.1007/s11856-017-1527-6

Abstract

We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form k(s,u)=∑ann−s−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 Hd2 in d variables, where d can be any number in {1, 2,...,∞}, and in particular their multiplier algebras are unitarily equivalent to the multiplier algebra of Hd2. Thus, a family of multiplier algebras of Dirichlet series is 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 as a reproducing kernel Hilbert space to Hd2 and when its multiplier algebra is isometrically isomorphic to Mult(Hd2).

Comments

This is a preprint version. The final publication is available at Springer via https://doi.org/10.1007/s11856-017-1527-6.

ORCID

http://orcid.org/0000-0003-0036-7606 [McCarthy)

Recommended Citation

McCarthy, John E. and Shalit, Orr Moshe, "Spaces of Dirichlet series with the complete Pick property" (2017). Mathematics Faculty Publications. 42.
https://openscholarship.wustl.edu/math_facpubs/42