SPACES OF DIRICHLET SERIES WITH THE COMPLETE PICK PROPERTY

We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form k(s, u) = Sigma a(n)n(-s-(u) over bar), 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, ... ,infinity}, 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)).