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

DOI:
https://doi.org/10.1090/S0002-9939-06-08303-1

Published electronically:
April 7, 2006

MathSciNet review:
2213750

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Abstract | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0002-9939-06-08303-1

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.