Operator Hölder–Zygmund functions

Abstract

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.

Multiple operator integrals in perturbation theory

“…Note that similar results hold for functions of unitary operators, contractions and dissipative operators, see [5,9].…”

“…3.7 that the divided differences D [1] f and D [2] f do not have to belong to the Haagerup tensor product L ∞ ⊗ h L ∞ ⊗ h L ∞ for an arbitrary function f in B 1 ∞,1 (R 2 ). The results of this sections were obtained in [4][5][6]. …”

“…In [5] it was shown that higher operator differences can be represented in terms of multiple operator integrals. This allowed one to obtain analogs of the results discussed in Chapter 1 for higher operator differences.…”

“…2) is necessarily operator Hölder of order α remained open for 40 years and it was solved in [4] (see also [5] for a detailed presentation). The solution is given by the following theorem:…”

Trace formulae for perturbations of class $\boldsymbol{{\boldsymbol S}_m}$

“…Strictly speaking, formula (6.1) was proved in [2] for bounded self-adjoint operators A. However, it is easy to see that the approximation procedure used in the proof of Lemma 5.2 in this paper also works to extend formula (6.1) to the case of unbounded A.…”