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

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 Sp, 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)||Sp ≤K||f||B1 ∞,1 max {||A1-A2||Sp, ||B1-B2||Sp, ||C1 - C2||Sp} 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. © 2017 American Mathematical Society.