Functions of triples of noncommuting self-adjoint operators under perturbations of class S_p

In this paper we study properties of functions of triples of not necessarily commuting self-adjoint operators. The main result of the paper shows that unlike in the case of functions of pairs of self-adjoint operators there is no Lipschitz type estimates in any Schatten-von Neumann norm S_p, 1 ≥ p ≥ ∞, for arbitrary functions in the Besov class B¹ _∞, ₁(ℝ³). In other words, we prove that for p ∈ [1,∞], there is no constant K > 0 such that the inequality ||f(A₁,B₁,C₁) - f(A₂,B₂,C₂)||S_p ≤K||f||B¹ _∞,1 max {||A₁-A₂||S_p, ||B₁-B₂||S_p, ||C₁ - C₂||S_p} holds for an arbitrary function f in B¹ _∞,1 (ℝ³) and for arbitrary finite rank self-adjoint operators A₁, B₁, C₁, A₂, B₂ and C₂.