Functions of operators on Hilbert spaces

希尔伯特空间上的算子函数

• 批准号: 0900870
• 负责人: Kenneth Dykema
• 金额: $ 9.21万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0900870&HistoricalAwards=false
• 关键词: Functions; operators; Hilbert; spaces

AbstractSkripkaThis award is funded under the American Recovery andReinvestment Act of 2009 (Public Law 111-5).The proposed project is on approximation of operator functions, whose prototype in the case of functions of a scalar argument is the Taylor polynomial approximation. Values of operator functions do not commute in general, which makes the analysis of such functions and, in particular, their approximations much subtler than in the classical case. Under certain assumptions, traces of the remainders of the first and second order approximations can be represented via spectral shift functions, which originate from Lifshits' work on the quantum theory of crystals in 1952. While higher order Taylor-type approximations are also of interest in applications (for instance, in perturbation theory for Schrodinger operators with long-range potentials), very little is known about the structure of their error terms. The project will concentrate on the study of the higher order Taylor-type approximations, in particular, on testing Koplienko's conjecture of 1984 on existence of higher order spectral shift measures.Perturbation theory has originated as mathematical modeling of some problems of quantum mechanics, where physical quantities are described by self-adjoint operators acting on a separable Hilbert space. The change of a value of an operator function under a perturbation of its argument is reflected in the spectral shift functions. A comprehensive theory with various applications, including those to perturbation theory for Schrodinger operators, scattering theory, and spectral flow, has been constructed for these functions. Finding higher order analogs of the spectral shift functions is one of the goals of the project. Many operators can be naturally affiliated with von Neumann algebras (for instance, the integrated density of states for some operators can be expressed in terms of the corresponding von Neumann algebras). We will work in both the original and the von Neumann algebra setting of the perturbation theory.

Abstractskripkathis奖是根据2009年的《美国复苏Andreinvestment Act》（公法111-5）资助的。拟议的项目是关于操作员函数的近似值，其原型在标量论点的函数方面是泰勒多项式近似。运算符函数的值一般不会通勤，这使得对此类功能的分析，尤其是它们的近似值比经典案例中的近似值要大得多。在某些假设下，可以通过光谱变化函数来表示剩余的剩余痕迹，该函数源自1952年Lifshits在晶体上的量子理论的工作。虽然高阶泰勒型近似值在应用中也具有兴趣（例如，在应用程序中，与schrodinginger operators for Schrodinger Operation相关的差异很少）。 The project will concentrate on the study of the higher order Taylor-type approximations, in particular, on testing Koplienko's conjecture of 1984 on existence of higher order spectral shift measures.Perturbation theory has originated as mathematical modeling of some problems of quantum mechanics, where physical quantities are described by self-adjoint operators acting on a separable Hilbert space.在其参数的扰动下，操作员函数值的变化反映在光谱变化函数中。具有各种应用的综合理论，包括针对施罗辛格运营商的扰动理论，散射理论和光谱流的扰动理论，已被构建为这些功能。找到光谱变化功能的高阶类似物是项目的目标之一。许多操作员可以自然隶属于von Neumann代数（例如，某些操作员的综合状态密度可以用相应的von Neumann代数表示）。我们将在扰动理论的原始和冯·诺伊曼代数设置中工作。