In Peller (1985, 1990) [10,11], Aleksandrov and Peller (2009, 2010, 2010) [1-3] sharp estimates for f(A)-f(B) were obtained for self-adjoint operators A and B and for various classes of functions f on the real line R. In this Note we extend those results to the case of functions of normal operators. We show that if f belongs to the Hölder class Λα(R2), 0 < α < 1, of functions of two variables, and N1 and N2 are normal operators, then {norm of matrix}f(N1)-f(N2){norm of matrix} ≤ const{norm of matrix}f{norm of matrix}Λα{norm of matrix}N1-N2{norm of matrix}α. We obtain a more general result for functions in the space Λω(R2)={f:|f(ζ1)-f (ζ2)| ≤ const ω(|ζ12|)} for an arbitrary modulus of continuity ω. We prove that if f belongs to the Besov class B∞11(R2), then it is operator Lipschitz, i.e., {norm of matrix}f(N1)-f(N2){norm of matrix} ≤ const{norm of matrix}f{norm of matrix}B∞11{norm of matrix}N1-N2{norm of matrix}. We also study properties of f(N1)-f(N2) in the case when f ∈ Λ α(R2) and N1-N2 belongs to the Schatten-von Neumann class Sp.

Translated title of the contributionFunctions of perturbed normal operators
Original languageFrench
Pages (from-to)553-558
Number of pages6
JournalComptes Rendus Mathematique
Volume348
Issue number9-10
DOIs
StatePublished - May 2010
Externally publishedYes

    Scopus subject areas

  • Mathematics(all)

ID: 87310809