Author's School

Arts & Sciences

Author's Department

Mathematics

Document Type

Article

Publication Date

8-2017

Originally Published In

Aleman, A., Hartz, M., McCarthy, J. E. and Richter, S. (2017), The Smirnov class for spaces with the complete Pick property. J. London Math. Soc., 96: 228–242. doi:10.1112/jlms.12060

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.

Comments

© 2017 London Mathematical Society

Author manuscript version of Aleman, A., Hartz, M., McCarthy, J. E. and Richter, S. (2017), The Smirnov class for spaces with the complete Pick property. J. London Math. Soc., 96: 228–242. doi:10.1112/jlms.12060

DOI

10.1112/jlms.12060

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