Triple operator integrals in Schatten--von Neumann norms and functions of perturbed noncommuting operators

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We study perturbations of functions f(A,B) of noncommuting self-adjoint operators A and B that can be defined in terms of double operator integrals. We prove that if f belongs to the Besov class B,1¹(²), then we have the following Lipschitz type estimate in the Schatten--von Neumann norm ₚ, 1≤ p≤2 norm: f(A₁,B₁)-f(A₂,B₂)ₚ≤(A₁-A₂ₚ+B₁-B₂ₚ). However, the condition f∈ B,1¹(²) does not imply the Lipschitz type estimate in ₚ with p>2. The main tool is Schatten--von Neumann norm estimates for triple operator integrals.