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

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