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

Richter和Sundberg将继续他们基于Agler关于算子族中的极值的思想的函数论算子理论和模型理论的研究。他们的工作重点之一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，作为理论和应用数学中广泛领域的相遇和思想来源，如复分析、调和分析、算子理论、泛函分析、控制论和偏微分方程。算子理论起源于19世纪末20世纪初弗雷德霍尔姆和希尔伯特关于偏微分方程的工作。解析函数空间的研究起源于二十世纪上半叶Hardy、Fischer和Riesz兄弟等人关于经典Hardy空间的工作。这两个领域在20世纪40年代S关于单边移位的工作中相遇，利用Hardy空间技巧产生了一个重要的无限维算子的完整结构理论。将Beurling的结果推广到任意多重数移位和Sz.Nagy膨胀定理是Hilbert空间上压缩算子模型理论的基础。因此，直到尺度，可分Hilbert空间上的每个有界线性算子都可以用向量值Hardy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运算符来建模，这具有在整个科学中都能感觉到的应用。需要进行更多的研究，并正在进行更多的研究，以澄清模型理论。特别是，已有大量的努力致力于将在Hardy空间的研究中形成的思想推广到解析函数的其他空间，特别是多变量设置。里希特和桑德伯格目前的工作就是在这一领域。

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.