Functions of perturbed pairs of non-commuting contractions

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.