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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A generalization of Andô's theorem and Parrott's example


Author: David Opela
Journal: Proc. Amer. Math. Soc. 134 (2006), 2703-2710
MSC (2000): Primary 47A20
Published electronically: April 7, 2006
MathSciNet review: 2213750
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Abstract: Andô's theorem states that any pair of commuting contractions on a Hilbert space can be dilated to a pair of commuting unitaries. Parrott presented an example showing that an analogous result does not hold for a triple of pairwise commuting contractions. We generalize both of these results as follows. Any $ n$-tuple of contractions that commute according to a graph without a cycle can be dilated to an $ n$-tuple of unitaries that commute according to that graph. Conversely, if the graph contains a cycle, we construct a counterexample.


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Additional Information

David Opela
Affiliation: Department of Mathematics, Campus Box 1146, Washington University in Saint Louis, Saint Louis, Missouri 63130
Email: opela@math.wustl.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08303-1
PII: S 0002-9939(06)08303-1
Keywords: Unitary dilations, commuting contractions, And\^o's theorem
Received by editor(s): January 2, 2005
Received by editor(s) in revised form: April 12, 2005
Published electronically: April 7, 2006
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.