Operator Theory and Function Theory for the unit ball of C^d

C^d 单位球的算子理论和函数理论

• 批准号: 0901642
• 负责人: Stefan Richter
• 金额: $ 26.77万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-10-01 至 2013-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901642&HistoricalAwards=false
• 关键词: Operator; Theory; Function; unit; ball

Richter and Sundberg will continue their research in Function-Theoretic Operator Theory and Model Theory based on Agler's idea of extremals in families of operators. One particular emphasis in their work will be a study of Hilbert modules based on the Drury-Arveson space of analytic functions in several complex variables.The study of spaces of analytic functions has a long and rich history as a meeting ground and source of ideas from a wide range of areas in both Pure and Applied Mathematics such as Complex Analysis, Harmonic Analysis, Operator Theory, Functional Analysis, Control theory, and Partial Differential Equations. Operator theory has its roots in the work on Partial Differential Equations of Fredholm and Hilbert in the late nineteenth and early twentieth centuries. The study of spaces of analytic functions has its origins in the work on the classical Hardy spaces by Hardy, Fischer, and the Riesz brothers, among others, in the first half of the twentieth century. The two areas met in the 1940's in the work of A. Beurling on the unilateral shift, which yielded a complete structure theory of an important infinite dimensional operator using Hardy-space techniques. The generalization of Beurling's results to arbitrary multiplicity shifts together with the Sz.Nagy dilation theorem is the basis for a model theory for contraction operators on Hilbert spaces. Thus, up to scaling, every bounded linear operator on a separable Hilbert space can be modeled using a semi-invariant subspace of a vector-valued Hardy space. As many naturally occurring processes can be modeled by use of such linear operators, this has applications that can be felt throughout science. More research is needed and is being done to clarify the model theory. In particular, there has been a large effort devoted to extending the ideas developed in the study of the Hardy spaces to other spaces of analytic functions, and in particular to the multivariable setting. The current work of Richter and Sundberg is in this area.

里希特和桑德伯格将继续他们的研究在功能理论算子理论和模型理论的基础上阿格勒的想法极值家庭的运营商。在他们的工作中，一个特别的重点将是基于多复变解析函数的Drury-Arveson空间的Hilbert模的研究。解析函数空间的研究有着悠久而丰富的历史，作为纯数学和应用数学中广泛领域的思想的交汇点和来源，如复分析，调和分析，算子理论，泛函分析，控制理论，偏微分方程算子理论起源于Fredholm和Hilbert在19世纪末和20世纪初对偏微分方程的研究。解析函数空间的研究起源于世纪上半叶哈代、费舍尔和Riesz兄弟对经典哈代空间的研究。这两个领域在20世纪40年代的A. Beurling的单边移位，这产生了一个完整的结构理论的一个重要的无限维运营商使用哈代空间技术。将Beurling的结果推广到任意重数移位以及Sz.Nagy膨胀定理是Hilbert空间上压缩算子的模型理论的基础。因此，在尺度上，可分希尔伯特空间上的每一个有界线性算子都可以用向量值哈代空间的半不变子空间来建模。由于许多自然发生的过程可以通过使用这样的线性算子来建模，这在整个科学中都有应用。需要更多的研究，并正在做澄清模型理论。特别是，一直有大量的努力致力于扩大思想发展的研究哈代空间的其他空间的解析函数，特别是多变量设置。Richter和Sundberg目前的工作就是在这个领域。