Inequality of von Neumann for more than two contractions

Question

Good morning,

I'm doing the Master 2 Practice at the University of Toulouse 3, France, on the spectral Nevanlinna-Pick interpolation, via operator theory. This problem leads to study the symmetrized polydisc $\mathbb{G}_n$ defined as follows

$$\mathbb{G}_n =\{(\sigma_1(\lambda),\ldots, \sigma_n(\lambda))~:~ \lambda\in \mathbb{D}^n\}$$ where $$\sigma_i(\lambda)=\sum_{1\leq j_1<j_2<\ldots<j_i\leq n} \lambda_{j_1}\lambda_{j_2}\ldots\lambda_{j_i}$$ are the elementary symmetric polynomials of $\lambda = (\lambda_1,\ldots,\lambda_n).$

Some authors study these symmetrized polydiscs via operator theory, e.g Jim Agler and Nicholas Young. The main tools they used are the commutant lifting theorem and the inequality of von Neumann. These two authors obtained the following result via this approach : the Caratheodory distance and Kobayashi distance are equal for the symmetrized bidisc $\mathbb{G}_2$. This result is surprising, because the symmetrized bidisc is not biholomorphic to a convex domain (due to Costara), and can not be exhausted by domains biholomorphic to convex domains (due to Edigarian).

However, it's impossible to study the symmetrized polydisc of higher dimension via the commutant lifting/von Neumann's inequality, since these two theorems fail for more than two contractions. I think understanding why these fail will give informations on the symmetrized polydisc and the spectral Nevanlinna-Pick interpolation problem.

Question : I would like to know if there are papers which gather the informations on the failure of the commutant lifting theorem and the inequality of von Neumann for more than two contractions.

Any help is appreciated. Thanks in advance.

Duc Anh

Answer 1

A good reference is Pisier's monograph Similarity problems and completely bounded maps. The first chapter is devoted to the von Neumann inequality and its generalizations to two and more contractions. There are also many references.

Answer 2

You might want to look at a recent paper by my student David Opela, "A Generalization of Ando's Theorem and Parrott's Example", arXiv:math/0505154v1. It "explains" why you can dilate a pair, but not a triple, of commuting contractions to unitaries.

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.

Answer 3

You may also want to look at the paper "Classes of tuples of commuting contractions satisfying the multivariable von Neumann inequality" by Grinshpan, Kaliuzhnyi-Verbovetskyi, Vinnikov, and Woerdeman (Journal of Functional Analysis, Volume 256, Issue 9, 1 May 2009, Pages 3035–3054). They give some additional sufficient conditions for commuting contractions to admit a unitary dilation.