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 d 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 d . 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 d and when its multiplier algebra is isometrically isomorphic to Mult(H 2 d ).
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Partially supported by National Science Foundation Grant DMS 1300280.
Partially supported by ISF Grant 474/12 and EU FP7/2007-2013 Grant 321749.
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McCarthy, J.E., Shalit, O.M. Spaces of Dirichlet series with the complete Pick property. Isr. J. Math. 220, 509–530 (2017). https://doi.org/10.1007/s11856-017-1527-6
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DOI: https://doi.org/10.1007/s11856-017-1527-6