Multivariable Operator Theory and Applications

一元全纯函数可以很容易地应用于谱在函数域内的矩阵，这已发展成为一个非常成功和重要的理论。首席研究员建议研究应用于矩阵的几个变量的函数。该理论立即分为对易和非对易两种情况。在前者中，该项目将研究多个变量的函数何时保持矩阵的自然顺序结构，以及当矩阵的大小变为无穷大时(算符情况)会发生什么。相反，首席调查者建议使用算子理论中的工具来理解多个变量的函数，例如刻画Pick类中函数的渐近展开(这称为单变量的Hamburger问题)。在非对易的情况下，他将研究如何获得非对易全纯函数作为非对易多项式的极限。非对易全纯函数的描述将以算子理论实现的方式进行。他还将致力于应用分析技术来改进超声成像。实际上，今天所有的成像设备都是通过收集电磁波或声波，并利用这些波携带的能量来确定像素值，以建立基本上是“能量”的图像。然而，波也携带信息，这也可以用来确定图像中的像素值。各种研究表明，这些“信息”图像往往揭示了传统的能量图像完全忽略的特征。这个项目将研究如何使用各种不同的信息论熵测量来改善这些图像，既使它们变得更清晰，又使它们能够实时计算。这些技术将在医学成像和安全扫描中得到应用。首席调查员还将在与控制理论和信号处理有关的纯数学领域进行基础研究。

A holomorphic function of a single variable can easily be applied to matrices whose spectra lie in the domain of the function, and this has developed into a very successful and important theory. The Principal Investigator proposes to study functions of several variables applied to matrices. The theory immediately splits into the commutative and noncommutative cases. In the former, the project will investigate when functions of several variables preserve the natural order structure on matrices, and what happens when the size of the matrices becomes infinite (the operator case). Conversely, the Principal Investigator proposes to use tools from operator theory to understand functions of several variables, such as characterizing asymptotic expansions of functions in the Pick class (this is called the Hamburger problem in one variable). In the noncommutative case, he will investigate how noncommutative holomorphic functions can be obtained as limits of noncommutative polynomials. The description of noncommutative holomorphic functions will be in terms of operator theoretic realizations. He will also work on applying analytic techniques to improve ultrasound imaging.Virtually all imaging devices today function by collecting either electromagnetic or acoustic waves and using the energy carried by these waves to determine pixel values in order to build up what is basically an "energy" picture. However, waves also carry information, and this can also be used to determine the pixel values in an image. Various studies have shown that these "information" images often reveal features that are completely missed by conventional, energy images. This project will study how to improve these images, using various different measures of information-theoretic entropy, both to render them sharper, and to enable their calculation in real time. These techniques will have applications in medical imaging and also in security scanning. The Principal Investigator will also conduct basic research in areas of pure mathematics related to control theory and signal processing.