Topics in Dilation Theory

提案：DMS-9970347主要研究者：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。一个共同的主题是膨胀的运营商和随之而来的提升交织地图-交换提升，因为它是所谓的-和特殊情况下插值的Nevanlinna-Pick和Caratheodory类型。Nevanlinna-Pick核，也称为NP核，是由Agler引入的Szego核的自然推广。例子包括狄利克雷核，多连通域上近似等于Szego核的核，以及Fock空间球上的核。膨胀和交换提升结果与这些内核的经典结果并行，包括两孔域上丛位移的交换提升。NP核和多连通域的一些问题仍然没有得到解决。多连通域上的所有函数论对象都可以用θ函数来描述。从三割线的身份，与知识相结合的零θ函数，它遵循任何给定的标量值Nevanlinna-Pick问题在一个环它足以考虑只有两个内核，这是明确确定的数据。Theta函数将被用来调查额外的问题，在环和更一般的多连通域。阿格勒算子族的极值元素（大致是代数学家所称的投射元素）是提升或扩张该算子族的一般成员的自然对象。极值是C*-闭的族，例如收缩族、等距族和次正规收缩族，从模型论的观点来看是特别好的。这不是亚正规压缩和数值半径压缩的情况，其中极值是族的真子集，但极值的C*-闭包是整个族。目前还没有一个很好的度量极值与其C*-闭包的发散性的方法。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一个有利的整体一致。例如，某些地震资料表明存在石油储量，而其他地震数据则与问题无关。类似地，在电路设计中，通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以一种很好的方式将给定的信息表示为一个更大、更连贯的整体的一部分。McCullough将研究将“局部”嵌入“整体”的新策略，并寻求统一许多不同应用的内聚理论。这将在被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驾驶仪等自动控制器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.