A generalization of Andô's theorem and Parrott's example
Author:
David Opela
Journal:
Proc. Amer. Math. Soc. 134 (2006), 27032710
MSC (2000):
Primary 47A20
Published electronically:
April 7, 2006
MathSciNet review:
2213750
Fulltext PDF Free Access
Abstract 
References 
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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 tuple of contractions that commute according to a graph without a cycle can be dilated to an 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/S0002993906083031
PII:
S 00029939(06)083031
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.
