Functions of noncommuting self-adjoint operators under perturbation and estimates of triple operator integrals

Abstract

We define functions of noncommuting self-adjoint operators with the help of double operator integrals. We are studying the problem to find conditions on a function $f$ on ${\mathbb{R}}^2$, for which the map $(A,B)\mapsto f(A,B)$ is Lipschitz in the operator norm and in Schatten--von Neumann norms ${\mathbf{S}}_p$. It turns out that for functions $f$ in the Besov class $B_{\infty ,1}^1({\mathbb{R}}^2)$, the above map is Lipschitz in the ${\mathbf{S}}_p$ norm for $p\in [1,2]$. However, it is not Lipschitz in the operator norm, nor in the ${\mathbf{S}}_p$ norm for $p>2$. The main tool is triple operator integrals. To obtain the results, we introduce new Haagerup-like tensor products of $L^{\infty }$ spaces and obtain Schatten--von Neumann norm estimates of triple operator integrals. We also obtain similar results for functions of noncommuting unitary operators.

43 pages