Topics in Dilation Theory

建议：DMS-9970347首席研究员：Scott A.McCullough摘要：McCullough计划研究泛函分析和算子理论中的几个相关问题。一个共同的主题是算子的扩张和交织映射的伴随提升--它被称为交换提升--以及Nevanlinna-Pick和Caratheodory类型的插补的特殊情况。Nevanlinna-Pick核，也称为NP核，是由Agler作为Szego核的自然推广而引入的。例子包括Dirichlet核，重连通域上几乎等于Szego核的核，以及Fock空间中球上的核。伸缩和对易提升结果平行经典结果对这些核成立，包括双孔区域上丛移位的对易提升。NP核和多重连通域的许多问题仍然没有解决。多连通域上的所有函数论对象都可以用theta函数来描述。从三分式恒等式出发，结合theta函数的零点的知识，得出对于环上任何给定的标量值Nevanlinna-Pick问题，只要考虑两个核就足够了，这两个核是由数据显式确定的。Theta函数将被用来研究环域和更一般的多重连通区域上的其他问题。Agler算子族的极值元素(大致就是代数学家所说的投射元素)是提升或扩展该算子族的一般成员的自然对象。从模型理论的观点来看，极点是C*-闭的族，如收缩、等距和次正常收缩的族，特别好。这不适用于非正规收缩，也不适用于数值半径收缩，其中极值是族的真子集，但极值的C*-闭包是整个族。目前还不存在很好的测量极值与其C*闭包的发散度的方法。在物理和工程中出现的一个常见的数学问题是确定某些部分信息是否与有利的整体一致。例如，某些地震学信息表明石油储量的存在，而其他地震数据与这个问题无关。同样，在电路设计中，经常有几个相互竞争的设计标准，人们必须从中选择相关的标准。这些问题也可以被视为以一种很好的方式将给定的信息表现为一个更大、更连贯的整体的一部分。麦卡洛将研究将“部分”嵌入“整体”的新策略，并寻求统一许多不同应用的凝聚力理论。这将在被称为算符理论的数学领域的框架内完成。其基本数学原理与数字技术和自动驾驶仪等自动控制器中使用的数学原理非常相似。

Proposal: DMS-9970347Principal Investigator: Scott A. McCulloughAbstract: McCullough plans to investigate several related problems in functional analysis and operator theory. A common theme is the dilation of operators and the concomitant lifting of intertwining maps -- commutant lifting, as it is called -- and the special cases of interpolation of Nevanlinna-Pick and Caratheodory type. Nevanlinna-Pick kernels, also known as NP kernels, were introduced by Agler as a natural generalization of the Szego kernel. Examples include the Dirichlet kernel, kernels nearly equal the Szego kernel on a multiply connected domain, and the kernels over the ball for Fock space. Dilation and commutant lifting results that parallel classical results hold for these kernels, including commutant lifting for bundle shifts over a two-holed domain. A number of issues for NP kernels and multiply connected domains remain unresolved. All function theoretic objects over a multiply connected domain can be described in terms of theta functions. From the trisecant identity, in combination with with knowledge of the zeros of theta functions, it follows that for any given scalar valued Nevanlinna-Pick problem over an annulus it suffices to consider just two kernels, which are explicitly determined from the data. Theta functions will be used to investigate additional problems over the annulus and more general multiply connected domains. The extremal elements (roughly what an algebraist would term projective elements) of an Agler family of operators are the natural objects to which to lift or dilate generic members of the family. Families for which the extremals are C*-closed, such as the families of contractions, isometries, and subnormal contractions, are particularly nice from a model theoretic viewpoint. This is not the case for the hyponormal contractions nor for the numerical radius contractions, where the extremals are a proper subset of the family but the C*-closure of the extremals is the entire family. A good measure of the divergence of the extremals from their C*-closure does not yet exist.A common mathematical problem that arises in physics and engineering is to determine whether certain partial information is consistent with a favorable whole. For instance, certain seismographic information indicates the presence of oil reserves, while other seismic data are not pertinent to the question. Similarly, in the design of an electrical circuit there are often several competing design criteria from which one must select the relevant ones. These problems can also be viewed as representing given information in a nice way as a part of a larger, more coherent whole. McCullough will investigate new stategies for imbedding the "partial" into the "whole," and conduct a search for a cohesive theory unifying many different applications. This will be done in the framework of the mathematical field known as operator theory. The underlying mathematics is quite similar to that used in digital technology and automatic controlers such as autopilots.