Spaces of Dirichlet series with the complete Pick property |
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Authors: | John E McCarthy Orr Moshe Shalit |
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Institution: | 1.Department of Mathematics,Washington University,St. Louis,USA;2.Faculty of Mathematics,Technion—Israel Institute of Technology,Haifa,Israel |
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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 d 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 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 ). |
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