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

2017/05/19 by Peller, V. V.
#Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1706.01969

Abstract

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 \boldsymbol S_p, 1≤ p≤∞, for arbitrary functions in the Besov class B_∞,1¹(\Bbb R³). In other words, we prove that for p∈[1,∞], there is no constant K>0 such that the inequality ‖f(A₁,B₁,C₁)amp;-f(A₂,B₂,C₂)‖_{\boldsymbol Sp}
amp;≤ K‖f‖_B_∞,1¹ max\‖A₁-A₂‖_{\boldsymbol Sp},‖B₁-B₂‖_{\boldsymbol Sp},‖C₁-C₂‖_{\boldsymbol Sp}\ holds for an arbitrary function f in B_∞,1¹(\Bbb R³) and for arbitrary finite rank self-adjoint operators A₁, B₁, C₁, A₂, B₂ and C₂.

Related

• Functions of triples of noncommuting self-adjoint operators and their perturbations
• Functions of noncommuting self-adjoint operators under perturbation and estimates of triple operator integrals
• Functions of noncommuting operators under perturbation of class \boldsymbolS_p
• Triple operator integrals in Schatten--von Neumann norms and functions\n of perturbed noncommuting operators
• Functions of perturbed noncommuting self-adjoint operators