Topics in Dilation Theory

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——即所谓的交换提升——以及Nevanlinna-Pick和Caratheodory类型的插值的特殊情况。Nevanlinna-Pick核，也被称为NP核，是Agler作为Szego核的自然推广引入的。例子包括Dirichlet核，在多连通域上的核几乎等于Szego核，以及Fock空间上球上的核。这些核的膨胀和交换子提升结果与经典结果相平行，包括双孔区域上束位移的交换子提升。NP核和多连通域的许多问题仍未得到解决。在一个多连通域上的所有函数理论对象都可以用函数来描述。从三等分恒等式，结合函数零点的知识，可以得出，对于环空上任何给定的标量值Nevanlinna-Pick问题，只要考虑两个核就足够了，这两个核是由数据明确确定的。函数将用于研究环空和更一般的多连通域上的附加问题。Agler算子族的极值元素（大致就是代数学家所说的射影元素）是用来提升或扩展该族的一般成员的自然对象。从模型理论的观点来看，极端是C*闭的族，如收缩族、等距族和次正规收缩族，是特别好的。对于次正规收缩和数值半径收缩，情况并非如此，其中极值是族的一个适当子集，但极值的C*闭包是整个族。极值与C*闭包之间的散度还没有一个很好的测量方法。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一个有利的整体一致。例如，某些地震资料表明存在石油储备，而其他地震资料则与这个问题无关。同样，在电路设计中，经常有几个相互竞争的设计标准，必须从中选择相关的标准。这些问题也可以被看作是一个更大、更连贯的整体的一部分，以一种良好的方式表示给定的信息。麦卡洛将研究将“局部”嵌入“整体”的新策略，并寻求一种统一许多不同应用的内聚理论。这将在称为算子理论的数学领域的框架内完成。其基础数学与数字技术和自动控制器（如自动驾驶仪）中使用的数学非常相似。

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.