Functions of triples of noncommuting self-adjoint operators under perturbations of class SpS_pSp​ (original) (raw)

2017, Proceedings of the American Mathematical Society

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 B 1 ∞,1 (R 3). 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 B 1 ∞,1 max A1 − A2 S p , B1 − B2 S p , C1 − C2 S p holds for an arbitrary function f in B 1 ∞,1 (R 3) and for arbitrary finite rank self-adjoint operators A1, B1, C1, A2, B2 and C2.