Operator Theory and Quadrature Formulas

主要研究人员：Mihai Putina该项目专注于三个主题，它们都相互联系得很好：通过秩一自交换子的次正规算子的调和函数的求积域，通过伸缩理论(具有指定的交换子，用于自伴算子的元组)用于固定次数多项式的多变量的体积公式，以及希尔伯特空间中的球面等距的谱图。第一个主题的基础是过去五年来用线性代数技术研究求积域的进展，特别是利用矩阵对的行列式函数。与经典的自伴算子微扰理论有许多相似之处，它们指向了由这种方法产生的几个公开问题。第二个主题是基于几年前P.I.的观察，即几乎所有与多元立方体有关的事实都可以翻译成熟悉的伸缩理论术语(交换雅可比矩阵)。预计这一观点将被双方成功利用。第三个主题是第二个主题的特化，即由奇数领域支持的积极措施。复变量和相关的Toeplitz算子(它们是球面等距的典型例子--现代算符理论中普遍存在的物体)被期望以封闭、灵活的形式携带关于原始测量的基本信息。逆问题，即从部分数据重建复杂对象的过程，在现代科学技术中随处可见。举几个经典的例子，我们可以提到弦的振动，知道其遥远引力场的行星的形状，或者水中漏油的演化。该提议旨在解决与形状或质量分布相关的反问题的一些数学问题。这一领域一个多世纪的非凡发现的历史将与现代数学的最新进展相结合。由此产生的工作将对目前数学领域的研究前沿以及潜在的应用领域，如地球物理、图像处理或层析成像具有重要意义。

Principal Investigator: Mihai Putinar The project focuses on three themes, all well connected to each other: quadrature domains for harmonic functions via hyponormal operators of rank-one self-commutator, cubature formulas in several variables for polynomials of a fixed degree via dilation theory (with prescribed commutators, for tuples of self-adjoint operators) and the spectral picture of spherical isometries in a Hilbert space. The basis for the first subject is the progress made in the last five years in the study of quadrature domains with linear algebra techniques, and specifically using determinantal functions of pairs of matrices. There are many analogies with classical perturbation theory of self-adjoint operators which point to several open problems resulting from this approach. The second theme is based on the observation made a couple of years ago by the P.I. that practically all facts connected with multivariate cubatures can be translated into familiar terms of dilation theory (of commuting Jacobi matrices). It is expected that this point of view will be successfully exploited by both sides. The third subject is a specialization of the second, for positive measures supported by odd spheres. The complex variables and the associated Toeplitz operators (which are typical examples of spherical isometries-- ubiquitous objects in modern operator theory) are expected to carry in a closed, flexible form the basic information about the original measure. Inverse problems, that is the reconstruction of complex objects from partial data, are everywhere present in modern science and technology. To give a few classical examples we can mention the vibrations of a string, the shape of a planet knowing its distant gravitation field, or the evolution of an oil spill in water. The proposal is aimed at solving some mathematical questions related to inverse problems related to shapes or to mass distributions. A history of more than one century of remarkable discoveries in this area will be comb ined with recent advances in modern mathematics. The resulting work will be significant for the present research frontier in mathematics and potentially for applied fields such as geophysics, image processing or tomography.