2009/07/17 by А. Б. Александров, Aleksandrov, A. B., Vladimir Peller +1 · 3 citations
Mathematics · #46E15 #46E35 #47A55 #47A60 #47B49 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.0907.3049
openalex publication_date 2009/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Ł_\a(\R) with 00. Then we find a sharp estimate for ‖f(A)-f(B)‖ for functions f of class Ł_ø\df\f: øf(\d)≤\constø(\d)\ for an arbitrary modulus of continuity ø. In particular, we study moduly of continuity, for which ‖f(A)-f(B)‖≤\constø(‖A-B‖) for self-adjoint A and B, and for an arbitrary function f in Ł_ø. 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(m j)f(A+jK) for f in the class Łø,m that is defined in terms of finite differences and a modulus continuity ø of order m. We also obtaine similar results for unitary operators and for contractions.