TY - JOUR

T1 - Functions of normal operators under perturbations

AU - Александров, Алексей Борисович

AU - Peller, V. V.

AU - Potapov, D. S.

AU - Sukochev, F. A.

N1 - Funding Information: Keywords: Normal operators; Operator Lipschitz functions; Hölder classes; Besov classes; Schatten–von Neumann classes; Perturbations; Modulus of continuity; Double operator integrals; Commutators ✩ The first author is partially supported by RFBR grant 08-01-00358-a and by Russian Federation presidential grant NSh-2409.2008.1; the second author is partially supported by NSF grant DMS 1001844 and by ARC grant. * Corresponding author. E-mail address: peller@math.msu.edu (V.V. Peller).

PY - 2011/4/1

Y1 - 2011/4/1

N2 - In Peller (1980) [27], Peller (1985) [28], Aleksandrov and Peller (2009) [2], Aleksandrov and Peller (2010) [3], and Aleksandrov and Peller (2010) [4] 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 paper we extend those results to the case of functions of normal operators. We show that if a function f belongs to the Hölder class Λα(R2), 0<α<1, of functions of two variables, and N1 and N2 are normal operators, then {double pipe}f(N1){double pipe}f(N2){double pipe}≤const{double pipe}f{double pipe}Λα{double pipe}N1-N2{double pipe}α. We obtain a more general result for functions in the space Λω(R2)={f:|f(ζ1)-f(ζ2)|≤constω(|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., {double pipe}f(N1)-f(N2){double pipe}≤const{double pipe}f{double pipe}B∞11{double pipe}N1-N2{double pipe}. We also study properties of f(N1)-f(N2) in the case when f∈Λα(R2) and N1-N2 belongs to the Schatten-von Neumann class Sp.

AB - In Peller (1980) [27], Peller (1985) [28], Aleksandrov and Peller (2009) [2], Aleksandrov and Peller (2010) [3], and Aleksandrov and Peller (2010) [4] 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 paper we extend those results to the case of functions of normal operators. We show that if a function f belongs to the Hölder class Λα(R2), 0<α<1, of functions of two variables, and N1 and N2 are normal operators, then {double pipe}f(N1){double pipe}f(N2){double pipe}≤const{double pipe}f{double pipe}Λα{double pipe}N1-N2{double pipe}α. We obtain a more general result for functions in the space Λω(R2)={f:|f(ζ1)-f(ζ2)|≤constω(|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., {double pipe}f(N1)-f(N2){double pipe}≤const{double pipe}f{double pipe}B∞11{double pipe}N1-N2{double pipe}. We also study properties of f(N1)-f(N2) in the case when f∈Λα(R2) and N1-N2 belongs to the Schatten-von Neumann class Sp.

KW - Besov classes

KW - Commutators

KW - Double operator integrals

KW - Hölder classes

KW - Modulus of continuity

KW - Normal operators

KW - Operator Lipschitz functions

KW - Perturbations

KW - Schatten-von Neumann classes

UR - http://www.scopus.com/inward/record.url?scp=79952039943&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2011.01.008

DO - 10.1016/j.aim.2011.01.008

M3 - Article

VL - 226

SP - 5216

EP - 5251

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 6

ER -