## Abstract

We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form k(s, u) = ∑ a_{n}n^{−} ^{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 H_{d} ^{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 H_{d} ^{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 H_{d} ^{2} and when its multiplier algebra is isometrically isomorphic to Mult(H_{d} ^{2}).

Original language | English |
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Pages (from-to) | 509-530 |

Number of pages | 22 |

Journal | Israel Journal of Mathematics |

Volume | 220 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 2017 |