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(|ζ1-ζ2|)} 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 contribution | Functions of perturbed normal operators |
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Original language | French |
Pages (from-to) | 553-558 |
Number of pages | 6 |
Journal | Comptes Rendus Mathematique |
Volume | 348 |
Issue number | 9-10 |
DOIs | |
State | Published - May 2010 |
Externally published | Yes |
ID: 87310809