TY - JOUR

T1 - Operator Hölder-Zygmund functions

AU - Александров, Алексей Борисович

AU - Peller, V. V.

N1 - Funding Information: * Corresponding author. E-mail address: peller@math.msu.edu (V.V. Peller). 1 Partially supported by RFBR grant 08-01-00358-a and by Russian Federation presidential grant NSh-2409.2008.1. 2 Partially supported by NSF grant DMS 0700995 and by ARC grant.

PY - 2010/6

Y1 - 2010/6

N2 - It is well known that a Lipschitz function on the real line does not have to be operator Lipschitz. We show that the situation changes dramatically if we pass to Hölder classes. Namely, we prove that if f belongs to the Hölder class Λα(R{double-struck}) with 0<α<1, then {norm of matrix}f(A)-f(B){norm of matrix}≤const{norm of matrix}A-B{norm of matrix}α for arbitrary self-adjoint operators A and B. We prove a similar result for functions f in the Zygmund class Λ1(R{double-struck}): for arbitrary self-adjoint operators A and K we have {norm of matrix}f(A-K)-2f(A)+f(A+K){norm of matrix}≤const{norm of matrix}K{norm of matrix}. We also obtain analogs of this result for all Hölder-Zygmund classes Λα(R{double-struck}), α>0. Then we find a sharp estimate for {norm of matrix}f(A)-f(B){norm of matrix} for functions f of class Λω=def{f:ωf(δ)≤constω(δ)} for an arbitrary modulus of continuity ω. In particular, we study moduli of continuity, for which {norm of matrix}f(A)-f(B){norm of matrix}≤constω({norm of matrix}A-B{norm of matrix}) for self-adjoint A and B, and for an arbitrary function f in Λω. We obtain similar estimates for commutators f(A)Q-Qf(A) and quasicommutators f(A)Q-Qf(B). Finally, we estimate the norms of finite differences ∑j=0m(-1)m-j(mj)f(A+jK) for f in the class Λω,m that is defined in terms of finite differences and a modulus continuity ω of order m. We also obtain similar results for unitary operators and for contractions.

AB - It is well known that a Lipschitz function on the real line does not have to be operator Lipschitz. We show that the situation changes dramatically if we pass to Hölder classes. Namely, we prove that if f belongs to the Hölder class Λα(R{double-struck}) with 0<α<1, then {norm of matrix}f(A)-f(B){norm of matrix}≤const{norm of matrix}A-B{norm of matrix}α for arbitrary self-adjoint operators A and B. We prove a similar result for functions f in the Zygmund class Λ1(R{double-struck}): for arbitrary self-adjoint operators A and K we have {norm of matrix}f(A-K)-2f(A)+f(A+K){norm of matrix}≤const{norm of matrix}K{norm of matrix}. We also obtain analogs of this result for all Hölder-Zygmund classes Λα(R{double-struck}), α>0. Then we find a sharp estimate for {norm of matrix}f(A)-f(B){norm of matrix} for functions f of class Λω=def{f:ωf(δ)≤constω(δ)} for an arbitrary modulus of continuity ω. In particular, we study moduli of continuity, for which {norm of matrix}f(A)-f(B){norm of matrix}≤constω({norm of matrix}A-B{norm of matrix}) for self-adjoint A and B, and for an arbitrary function f in Λω. We obtain similar estimates for commutators f(A)Q-Qf(A) and quasicommutators f(A)Q-Qf(B). Finally, we estimate the norms of finite differences ∑j=0m(-1)m-j(mj)f(A+jK) for f in the class Λω,m that is defined in terms of finite differences and a modulus continuity ω of order m. We also obtain similar results for unitary operators and for contractions.

KW - Contractions

KW - Hölder classes

KW - Multiple operator integrals

KW - Operator Hölder functions

KW - Operator Lipschitz function

KW - Self-adjoint operators

KW - Unitary operators

KW - Zygmund class

UR - http://www.scopus.com/inward/record.url?scp=77952670582&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2009.12.018

DO - 10.1016/j.aim.2009.12.018

M3 - Article

VL - 224

SP - 910

EP - 966

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 3

ER -