Spaces of Dirichlet series with the complete Pick property [article]

John E. McCarthy, Orr Shalit
2015 arXiv   pre-print
We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form k(s,u) = ∑ a_n n^-s-u̅, 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_d in d variables, where d can be any number in {1,2,...,
more » ... }, and in particular their multiplier algebras are unitarily equivalent to the multiplier algebra of H^2_d. Thus, a family of multiplier algebras of Dirichlet series are 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 to H^2_d and when its multiplier algebra is isometrically isomorphic to Mult(H^2_d).
arXiv:1507.04162v2 fatcat:hwgw2i2frzenbd2jmx5pb42zxu