Functions of normal operators under perturbations

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• 作者：A.B. Aleksandrov^a ; V.V. Peller^b ; ^{peller@math.msu.edu} ; D.S. Potapov^c ; F.A. Sukochev^c
• 关键词：Normal operators ; Operator Lipschitz functions ; Hö ; lder classes ; Besov classes ; Schatten&#x2013 ; von Neumann classes ; Perturbations ; Modulus of continuity ; Double operator integrals ; Commutators
• 刊名：Advances in Mathematics
• 出版年：2011
• 出版时间：1 April 2011
• 年：2011
• 卷：226
• 期：6
• 页码：5216-5251
• 全文大小：275 K

文摘

In Peller (1980) [27], Peller (1985) [28], Aleksandrov and Peller (2009) [2], Aleksandrov and Peller (2010) [3], and Aleksandrov and Peller (2010) [4] 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 . 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 , 0<α<1, of functions of two variables, and N₁ and N₂ are normal operators, then f(N₁)−f(N₂)constf_ΛαN₁−N₂^α. We obtain a more general result for functions in the space for an arbitrary modulus of continuity ω. We prove that if f belongs to the Besov class , then it is operator Lipschitz, i.e., . We also study properties of f(N₁)−f(N₂) in the case when and N₁−N₂ belongs to the Schatten&#x2013;von Neumann class S_p.