The Smirnov class for spaces with the complete Pick property

Abstract

We show that every function in a reproducing kernel Hilbert space with a normalized complete Pick kernel is the quotient of a multiplier and a cyclic multiplier. This extends a theorem of Alpay, Bolotnikov and Kaptanoğlu. We explore various consequences of this result regarding zero sets, spaces on compact sets and Gleason parts. In particular, using a construction of Salas, we exhibit a rotationally invariant complete Pick space of analytic functions on the unit disc for which the corona theorem fails.

18 pages; minor changes

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• Gleason-Kahane-Żelazko theorems in function spaces
• Totally null sets and capacity in Dirichlet type spaces
• Sub-Bergman Hilbert spaces on the unit disk III
• Shift-cyclicity in analytic function spaces
• Which de Branges-Rovnyak spaces have complete Nevanlinna-Pick property?
• Embedding dimension of the Dirichlet space
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