Spaces of Dirichlet series with the complete Pick property

Abstract

We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form $k(s,u)=\sum a_n{n}^{-s-\bar{u}}$, 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 $\{1,2,\dots ,\infty \}$, 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 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 $H_d^2$ and when its multiplier algebra is isometrically isomorphic to $Mult(H_d^2)$.

23 pages

References in corpus (3)

• On the isomorphism problem for multiplier algebras of Nevanlinna-Pick spaces
• Multipliers of embedded discs
• Nevanlinna-Pick Kernels and Localization

Cited by in corpus (9)

• Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball
• On the isomorphism problem for multiplier algebras of Nevanlinna-Pick spaces
• The Smirnov class for spaces with the complete Pick property
• Interpolating sequences in spaces with the complete Pick property
• Algebras of noncommutative functions on subvarieties of the noncommutative ball: the bounded and completely bounded isomorphism problem
• Cyclicity preserving operators on spaces of analytic functions in ${\mathbb{C}}^n$
• The classification problem for operator algebraic varieties and their multiplier algebras
• Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball
• A Gleason-Kahane-Żelazko theorem for reproducing kernel Hilbert spaces