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 Λ α (R) with 0 < α < 1, then f (A) − f (B) const A − B α for arbitrary self-adjoint operators A and B. We prove a similar result for functions f in the Zygmund class Λ 1 (R): for arbitrary self-adjoint operators A and K we haveconst K . We also obtain analogs of this result for all Hölder-Zygmund classes Λ α (R), α > 0. Then we find a sharp estimate for f (A) − f (B) for functions f of class Λ ω def = {f : ω f (δ) const ω(δ)} for an arbitrary modulus of continuity ω. In particular, we study moduli of continuity, for which f (A) − f (B)const ω( A − B ) for self-adjoint A and B, and for an arbitrary function f in Λ ω . 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 m j =0 (−1) m−j m j f (A + jK) for f in the class Λ ω,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.

This is a continuation of our paper [2]. We prove that for functions f in the Hölder class Λ α (R) and 1 < p < ∞, the operator f (A) − f (B) belongs to S p/α , whenever A and B are self-adjoint operators with A − B ∈ S p . We also obtain sharp estimates for the Schatten-von Neumann norms f (A) − f (B) S p/α in terms of A − B S p and establish similar results for other operator ideals. We also estimate Schattenvon Neumann norms of higher order differences m j =0 (−1) m−j m j f (A + jK). We prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on f for f (A) − f (B) to belong to S q under the assumption that A − B ∈ S p . We also obtain Schatten-von Neumann estimates for quasicommutators f (A)R − Rf (B), and introduce a spectral shift function and find a trace formula for operators of the form f (A − K) − 2f (A) + f (A + K).

The purpose of this survey is a comprehensive study of operator Lipschitz functions. A continuous function f on the real line R os called operator Lipschitz if f (A) − f (B) ≤ const A − B for arbitrary self-adjoint operators A and B. We give sufficient conditions and necessary conditions for operator Lipschitzness. We also study the class of operator differentiable functions on R. Next, we consider operator Lipschitz functions on closed subsets of the plane and introduce the class of commutator Lipschitz functions on such subsets. An important role for the study of such classes of functions is played by double operator integrals and Schur multipliers. 3.8. A sufficient condition for commutator Lipschitzness in terms of Cauchy integrals 69 3.9. Commutator Lipschitz functions on the disk and on the half-plane 70 3.10. Operator Lipschitz functions and linear-fractional transformations 73 3.11. The spaces OL(R) and OL(T) 78 3.12. The spaces (OL) ′ (R) and (OL) ′ loc (T) 80 3.13. Around the sufficient condition by Arazy-Barton-Froedman 83 3.14. In which case does the equality OL(F) = Lip(F) holds? 91 Concluding remarks 92 References 94

In [Pe1], [Pe2], [AP1], [AP2], and [AP3] 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(R 2 ), 0 < α < 1, of functions of two variables, and N1 and N2 are normal operators, thenWe obtain a more general result for functions in the space Λω(R 2 ) = 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 classWe also study properties of f (N1) − f (N2) in the case when f ∈ Λα(R 2 ) and N1 − N2 belongs to the Schatten-von Neuman class Sp.

Weighted estimates are obtained for the derivatives in the model (shift-coinvariant) subspaces K p Θ , generated by meromorphic inner functions Θ of the Hardy class H p (C +). It is shown that the differentiation operator acts from K p Θ to a space L p (w), where the weight w depends on the function |Θ |, the rate of growth of the argument of Θ along the real line. As an application of the weighted Bernstein-type inequalities, new Carleson-type theorems on embeddings of the subspaces K p Θ in L p (µ) are proved. Also, results on the compactness of such embeddings are obtained, and properties of measures µ for which the norms · L p (µ) and · p are equivalent on a given model subspace K p Θ , are established.