Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование
Operator Hölder-Zygmund functions. / Александров, Алексей Борисович; Peller, V. V.
в: Advances in Mathematics, Том 224, № 3, 06.2010, стр. 910-966.Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование
}
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 -
ID: 5209492