TY - JOUR

T1 - Fonctions d'opérateurs perturbés normaux

AU - Александров, Алексей Борисович

AU - Peller, Vladimir

AU - Potapov, Denis

AU - Sukochev, Fedor

PY - 2010/5

Y1 - 2010/5

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77952675577&partnerID=8YFLogxK

U2 - 10.1016/j.crma.2010.04.015

DO - 10.1016/j.crma.2010.04.015

M3 - статья

AN - SCOPUS:77952675577

VL - 348

SP - 553

EP - 558

JO - Comptes Rendus Mathematique

JF - Comptes Rendus Mathematique

SN - 1631-073X

IS - 9-10

ER -