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An Agler-type model theorem for C0-semigroups of Hilbert space contractions

Rydhe, Eskil LU (2016) In Journal of the London Mathematical Society 93(2). p.420-438
Abstract
We investigate suitable conditions for a C0-semigroup (T(t))t≥0 of Hilbert space contractions to be unitarily equivalent to the restriction of the adjoint shift semigroup (S∗γ(t))t≥0 to an invariant subspace of the standard weighted Bergman space Aγ−2(C+,K). It turns out that (T(t))t≥0 admits a model by (S∗γ(t))t≥0 if and only if its cogenerator is γ-hypercontractive and limt→0T(t)=0 in strong operator topology. We then discuss how such semigroups can be characterized without involving the cogenerator. A sufficient condition is that, for each t>0, the operator T(t) is γ-hypercontractive. Surprisingly, this condition is necessary if and only if γ is integer. The paper is concluded with a conjecture that would imply a more symmetric... (More)
We investigate suitable conditions for a C0-semigroup (T(t))t≥0 of Hilbert space contractions to be unitarily equivalent to the restriction of the adjoint shift semigroup (S∗γ(t))t≥0 to an invariant subspace of the standard weighted Bergman space Aγ−2(C+,K). It turns out that (T(t))t≥0 admits a model by (S∗γ(t))t≥0 if and only if its cogenerator is γ-hypercontractive and limt→0T(t)=0 in strong operator topology. We then discuss how such semigroups can be characterized without involving the cogenerator. A sufficient condition is that, for each t>0, the operator T(t) is γ-hypercontractive. Surprisingly, this condition is necessary if and only if γ is integer. The paper is concluded with a conjecture that would imply a more symmetric characterization. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Journal of the London Mathematical Society
volume
93
issue
2
pages
19 pages
publisher
Oxford University Press
external identifiers
  • WOS:000374188000008
  • Scopus:84971602173
ISSN
0024-6107
DOI
10.1112/jlms/jdv067
language
English
LU publication?
yes
id
7a471588-76dd-4652-80e6-e465ac0ff15f
alternative location
http://jlms.oxfordjournals.org/cgi/content/full/jdv067? ijkey=NWdeYquUguXQG50&keytype=ref
date added to LUP
2016-04-19 09:32:39
date last changed
2017-02-06 11:27:56
@article{7a471588-76dd-4652-80e6-e465ac0ff15f,
  abstract     = {We investigate suitable conditions for a C0-semigroup (T(t))t≥0 of Hilbert space contractions to be unitarily equivalent to the restriction of the adjoint shift semigroup (S∗γ(t))t≥0 to an invariant subspace of the standard weighted Bergman space Aγ−2(C+,K). It turns out that (T(t))t≥0 admits a model by (S∗γ(t))t≥0 if and only if its cogenerator is γ-hypercontractive and limt→0T(t)=0 in strong operator topology. We then discuss how such semigroups can be characterized without involving the cogenerator. A sufficient condition is that, for each t>0, the operator T(t) is γ-hypercontractive. Surprisingly, this condition is necessary if and only if γ is integer. The paper is concluded with a conjecture that would imply a more symmetric characterization. },
  author       = {Rydhe, Eskil},
  issn         = {0024-6107},
  language     = {eng},
  number       = {2},
  pages        = {420--438},
  publisher    = {Oxford University Press},
  series       = {Journal of the London Mathematical Society},
  title        = {An Agler-type model theorem for <em style="margin: 0px; padding: 0px; border: 0px; outline-style: none; font-size: inherit; font-family: inherit; line-height: inherit; text-align: inherit; vertical-align: baseline;">C</em><sub style="margin: 0px; padding: 0px; border: 0px; outline-style: none; font-style: inherit; font-size: 0.85em; font-family: inherit; line-height: 0; text-align: inherit;">0</sub>-semigroups of Hilbert space contractions},
  url          = {http://dx.doi.org/10.1112/jlms/jdv067},
  volume       = {93},
  year         = {2016},
}