Functions of operators on Hilbert spaces

【摘要】skripka本奖项是根据2009年美国复苏和投资法案（公法111-5）资助的。本课题是关于算子函数的近似，在标量参数函数的情况下算子函数的原型是泰勒多项式近似。一般来说，算子函数的值不会交换，这使得对这类函数的分析，特别是它们的近似比经典情况要微妙得多。在某些假设下，一阶和二阶近似余数的痕迹可以通过谱移函数来表示，谱移函数起源于1952年Lifshits关于晶体量子理论的工作。虽然高阶泰勒型近似在应用中也很有意义（例如，在具有长程势的薛定谔算子的摄动理论中），但对其误差项的结构知之甚少。该项目将集中研究高阶泰勒型近似，特别是检验Koplienko 1984年关于存在高阶谱移测度的猜想。微扰理论起源于一些量子力学问题的数学建模，其中物理量是由作用于可分离希尔伯特空间的自伴随算子来描述的。算子函数在其参数扰动下值的变化反映在谱移函数中。为这些函数建立了一个具有多种应用的综合理论，包括薛定谔算子的微扰理论、散射理论和谱流。寻找谱移函数的高阶类似物是该项目的目标之一。许多算符可以自然地与冯·诺伊曼代数相关联（例如，一些算符的状态的积分密度可以用相应的冯·诺伊曼代数来表示）。我们将在原始的和冯诺依曼代数的摄动理论中进行研究。

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.