The Smirnov class for spaces with the complete Pick property
(2017) In Journal of the London Mathematical Society 96(1). p.228-242- 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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https://lup.lub.lu.se/record/46128b58-6793-402e-9008-fd986fb26305
- author
- Aleman, Alexandru LU ; Hartz, Michael ; McCarthy, John E. and Richter, Stefan
- organization
- publishing date
- 2017-08
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Journal of the London Mathematical Society
- volume
- 96
- issue
- 1
- pages
- 228 - 242
- publisher
- Oxford University Press
- external identifiers
-
- scopus:85021801775
- wos:000415345500012
- ISSN
- 0024-6107
- DOI
- 10.1112/jlms.12060
- language
- English
- LU publication?
- yes
- id
- 46128b58-6793-402e-9008-fd986fb26305
- date added to LUP
- 2017-07-20 09:37:01
- date last changed
- 2025-02-03 19:48:48
@article{46128b58-6793-402e-9008-fd986fb26305, abstract = {{<p>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.</p>}}, author = {{Aleman, Alexandru and Hartz, Michael and McCarthy, John E. and Richter, Stefan}}, issn = {{0024-6107}}, language = {{eng}}, number = {{1}}, pages = {{228--242}}, publisher = {{Oxford University Press}}, series = {{Journal of the London Mathematical Society}}, title = {{The Smirnov class for spaces with the complete Pick property}}, url = {{http://dx.doi.org/10.1112/jlms.12060}}, doi = {{10.1112/jlms.12060}}, volume = {{96}}, year = {{2017}}, }