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arXiv · 1701.07476

The Smirnov class for spaces with the complete Pick property

Alexandru Aleman·Michael Hartz·John E. McCarthy·Stefan Richter
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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.

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Alexandru Aleman, Michael Hartz, John E. McCarthy, Stefan Richter. 2017-07-26. The Smirnov class for spaces with the complete Pick property. https://doi.org/10.1112/jlms.12060

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