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Nevanlinna–Pick Interpolation and Factorization of Linear Functionals

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Abstract

If \({\mathfrak{A}}\) is a unital weak-* closed algebra of multiplication operators on a reproducing kernel Hilbert space which has the property \({\mathbb{A}_1(1)}\), then the cyclic invariant subspaces index a Nevanlinna–Pick family of kernels. This yields an NP interpolation theorem for a wide class of algebras. In particular, it applies to many function spaces over the unit disk including Bergman space. We also show that the multiplier algebra of a complete NP space has \({\mathbb{A}_1(1)}\), and thus this result applies to all of its subalgebras. A matrix version of this result is also established. It applies, in particular, to all unital weak-* closed subalgebras of H ∞ acting on Hardy space or on Bergman space.

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L-3G1, Canada

    Kenneth R. Davidson & Ryan Hamilton

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  1. Kenneth R. Davidson
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  2. Ryan Hamilton
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Correspondence to Ryan Hamilton.

Additional information

K.R. Davidson was partially supported by an NSERC grant.

R. Hamilton was partially supported by an NSERC fellowship.

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Davidson, K.R., Hamilton, R. Nevanlinna–Pick Interpolation and Factorization of Linear Functionals. Integr. Equ. Oper. Theory 70, 125–149 (2011). https://doi.org/10.1007/s00020-011-1862-7

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  • DOI: https://doi.org/10.1007/s00020-011-1862-7

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