A Generalization of Ando's Theorem and Parrott's Example
Operator Algebras
2007-05-23 v1 Functional Analysis
Abstract
Ando'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.
Cite
@article{arxiv.math/0505154,
title = {A Generalization of Ando's Theorem and Parrott's Example},
author = {David Opela},
journal= {arXiv preprint arXiv:math/0505154},
year = {2007}
}
Comments
6 pages, accepted in PAMS