Spectral shift function of higher order

• 成果类型: Article
• 署名作者: Potapov, Denis; Skripka, Anna; Sukochev, Fedor
• 署名单位: University of New South Wales Sydney; University of New Mexico
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-012-0431-2
• 发表日期: 2013
• 页码: 501-538
• 关键词: operator integrals

摘要

This paper resolves affirmatively Koplienko's (Sib. Mat. Zh. 25:62-71, 1984) conjecture on existence of higher order spectral shift measures. Moreover, the paper establishes absolute continuity of these measures and, thus, existence of the higher order spectral shift functions. A spectral shift function of order naacenter dot is the function eta (n) =eta (n,H,V) such that for every sufficiently smooth function f, where H is a self-adjoint operator defined in a separable Hilbert space ai and V is a self-adjoint operator in the n-th Schatten-von Neumann ideal S (n) . Existence and summability of eta (1) and eta (2) were established by Krein (Mat. Sb. 33:597-626, 1953) and Koplienko (Sib. Mat. Zh. 25:62-71, 1984), respectively, whereas for n > 2 the problem was unresolved. We show that eta (n,H,V) exists, integrable, and for some constant c (n) depending only on naacenter dot. Our results for eta (n) rely on estimates for multiple operator integrals obtained in this paper. Our method also applies to the general semi-finite von Neumann algebra setting of the perturbation theory.