Operator Hölder-Zygmund functions |
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Authors: | AB Aleksandrov VV Peller |
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Institution: | a St-Petersburg Branch, Steklov Institute of Mathematics, Fontanka 27, 191023 St-Petersburg, Russia b Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA |
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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 Λα(R) with 0<α<1, then for arbitrary self-adjoint operators A and B. We prove a similar result for functions f in the Zygmund class Λ1(R): for arbitrary self-adjoint operators A and K we have . We also obtain analogs of this result for all Hölder-Zygmund classes Λα(R), α>0. Then we find a sharp estimate for ‖f(A)−f(B)‖ for functions f of class for an arbitrary modulus of continuity ω. In particular, we study moduli of continuity, for which 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 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. |
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Keywords: | Operator Lipschitz function Operator Hö lder functions Self-adjoint operators Unitary operators Contractions Multiple operator integrals Hö lder classes Zygmund class |
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