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The Smirnov class for spaces with the complete Pick property

Aleman, Alexandru LU ; Hartz, Michael; McCarthy, John E. and Richter, Stefan (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.

Please use this url to cite or link to this publication:
author
organization
publishing date
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
2018-04-15 04:45:55
@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},
  volume       = {96},
  year         = {2017},
}