Functions of perturbed pairs of noncommuting contractions

Abstract

We consider functions $f(T,R)$ of pairs of noncommuting contractions on Hilbert space and study the problem for which functions $f$ we have Lipschitz type estimates in Schatten--von Neumann norms. We prove that if $f$ belongs to the Besov class $(B_{\infty ,1}^1{)}_{+}({\mathbb{D}}^2)$ of analytic functions in the bidisk, 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 the Besov space $(B_{\infty ,1}^1{)}_{+}({\mathbb{D}}^2)$, there are no Lipschitz such type estimates for $p>2$ as well as in the operator norm.

24 pages