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

Opěla, David

doi:10.1090/S0002-9939-06-08303-1

A generalization of Andô’s theorem and Parrott’s example
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Proc. Amer. Math. Soc. 134 (2006), 2703-2710 Request permission

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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