Operator Theory and Quadrature Formulas

该项目主要关注三个主题，它们彼此之间有很好的联系：通过秩一自对易子的次正规算子求调和函数的正交域，通过膨胀理论（对于自伴随算子的元组，使用规定的对易子）求固定次多项式的几个变量的立方体公式，以及希尔伯特空间中球面等距的谱图。第一个主题的基础是近五年来用线性代数技术研究正交域，特别是使用矩阵对的行列式函数所取得的进展。它与经典自伴随算子的微扰理论有许多相似之处，这些相似之处指出了由这种方法引起的几个开放问题。第二个主题是基于几年前P.I.的观察，即几乎所有与多元文化相关的事实都可以转化为膨胀理论（交换雅可比矩阵）的熟悉术语。预计双方将成功地利用这一观点。第三个主题是第二个主题的专门化，因为有奇数领域支持的积极措施。复变量和相关的Toeplitz算符（它们是球面等距的典型例子——现代算符理论中无处不在的对象）被期望以一种封闭的、灵活的形式携带有关原始度量的基本信息。逆问题，即从部分数据重构复杂对象，在现代科学技术中无处不在。举几个经典的例子，我们可以提到弦的振动，行星的形状知道它遥远的引力场，或者石油泄漏在水中的演变。该建议旨在解决一些与形状或质量分布相关的反问题相关的数学问题。一个多世纪以来这一领域的重大发现的历史将与现代数学的最新进展结合起来。由此产生的工作将对当前的数学研究前沿和潜在的应用领域，如地球物理学，图像处理或断层扫描具有重要意义。

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.