Operator Holder-Zygmund functions

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 Holder classes. Namely, we prove that if f belongs to the Holder class Lambda(alpha) (R) with 0 < alpha < 1, then parallel to f(A) - f (B)parallel to <= const parallel to A - B parallel to(alpha) for arbitrary self-adjoint operators A and B. We prove a similar result for functions f in the Zygmund class Lambda(1) (R): for arbitrary self-adjoint operators A and K we have parallel to f(A - K) - 2 f(A) + f(A + K)parallel to <= const parallel to K parallel to. We also obtain analogs of this result for all Holder-Zygmund classes Lambda(alpha)(R), alpha > 0. Then we find a sharp estimate for parallel to f(A) - f(B)parallel to for functions f of class Lambda(omega) =(def) {f : omega f(delta) <= const omega(delta)} for an arbitrary modulus of continuity omega. In particular, we study moduli of continuity, for which parallel to f(A) - f(B)parallel to <= const omega (parallel to A - B parallel to) for self-adjoint A and B, and for an arbitrary function f in A. 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 Sigma(m)(j=0)(-1)(m-j)((m)(j))f(A + jK) for f in the class Lambda(omega,m) that is defined in terms of finite differences and a modulus continuity omega of order in. We also obtain similar results for unitary operators and for contractions. (C) 2010 Elsevier Inc. All rights reserved.