Functions of perturbed pairs of non-commuting contractions

A. B. Aleksandrov ; V. V. Peller

Izvestiya. Mathematics , Tome 84 (2020) no. 4, pp. 659-682

Résumé

We consider functions $f(T,R)$ of pairs of noncommuting contractions on Hilbert space and study the problem as to which functions $f$ we have Lipschitz type estimates in Schatten–von Neumann norms. We prove that if $f$ belongs to the Besov class $(B_{\infty,1}^1)_+(\mathbb{T}^2)$ of analytic functions in the bidisc, then we have a Lipschitz type estimate for functions $f(T,R)$ of pairs of not necessarily commuting contractions $(T,R)$ in the Schatten–von Neumann norms $\mathbf{S}_p$ for $p\in[1,2]$. On the other hand, we show that for functions in $(B_{\infty,1}^1)_+(\mathbb{T}^2)$, there are no such Lipschitz type estimates for $p>2$, nor in the operator norm.

Détail

Keywords: semi-spectral measures, Schatten–von Neumann classes, double operator integrals, triple operator integrals, Haagerup tensor products, Haagerup-like tensor products
Mots-clés : contractions, perturbation, Besov classes.