# Inequality of von Neumann for more than two contractions

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

• This is not directly relevant (which is why I leave it as a comment) but via Nicholas Young's webpage www1.maths.leeds.ac.uk/~nicholas one can find a copy of the 1999 PhD thesis of David Ogle, which studies function/operator theory on the higher-dimensional symmetrized polydiscs. It doesn't discuss issues related to the failure of commutant liftig for $n>2$, but it might have some useful background. – Yemon Choi May 4 '12 at 0:01
• Thank you. In fact I've browsed his thesis, and this leads partially to my question. – Đức Anh May 4 '12 at 8:40