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