Functions of perturbed pairs of noncommuting contractions

TL;DR

This paper investigates Lipschitz estimates for functions of pairs of noncommuting contractions on Hilbert space, establishing results for certain Besov class functions within Schatten norms and identifying limitations outside those conditions.

Contribution

It proves Lipschitz estimates for functions in a specific Besov class on pairs of contractions within Schatten norms for p in [1,2], and shows these estimates do not hold for p>2 or in operator norm.

Findings

Lipschitz estimates hold for p in [1,2] within Schatten norms.

No Lipschitz estimates for p>2 or in the operator norm.

Identifies function class (Besov space) where estimates are valid.

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 $(B_{\infty,1}^1)_+({\Bbb D}^2)$ 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 $\boldsymbol{S}_p$ for $p\in[1,2]$. On the other hand, we show that for functions in the Besov space $(B_{\infty,1}^1)_+({\Bbb D}^2)$, there are no Lipschitz such type estimates for $p>2$ as well as in the operator norm.