A generalization of Andô’s theorem and Parrott’s example
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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.References
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Additional Information
- David Opěla
- Affiliation: Department of Mathematics, Campus Box 1146, Washington University in Saint Louis, Saint Louis, Missouri 63130
- Email: opela@math.wustl.edu
- 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
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 2703-2710
- MSC (2000): Primary 47A20
- DOI: https://doi.org/10.1090/S0002-9939-06-08303-1
- MathSciNet review: 2213750