Spaces of Dirichlet series with the complete Pick property

John E. McCarthy, Orr Moshe Shalit

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form k(s, u) = ∑ ann s , 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 Hd 2 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 Hd 2. 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 Hd 2 and when its multiplier algebra is isometrically isomorphic to Mult(Hd 2).

Original languageEnglish
Pages (from-to)509-530
Number of pages22
JournalIsrael Journal of Mathematics
Issue number2
StatePublished - Jun 1 2017


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