Spaces of Dirichlet series with the complete Pick property

Abstract

We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form $$k\left( {s,u} \right) = \sum {{a_n}} {n^{ - s - \overline u }}$$ , 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 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 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 2 and when its multiplier algebra is isometrically isomorphic to Mult(H 2 ).