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

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 R2R2, for which the map (A,B)↦f(A,B)(A,B)↦f(A,B) is Lipschitz in the operator norm and in Schatten–von Neumann norms SpSp. It turns out that for functions f   in the Besov class B∞,11(R2), the above map is Lipschitz in the SpSp norm for p∈[1,2]p∈[1,2]. However, it is not Lipschitz in the operator norm, nor in the SpSp norm for p>2p>2. The main tool is triple operator integrals. To obtain the results, we introduce new Haagerup-like tensor products of L∞L∞ spaces and obtain Schatten–von Neumann norm estimates of triple operator integrals. We also obtain similar results for functions of noncommuting unitary operators.