Dilation Theory, Non-commutative Convexity and Systems

这个建议的研究是在功能分析和运营商理论有关的工程系统理论和多变量和多连通域在复平面。 线性控制理论中的许多问题可以用多个非交换变量的多项式来表示。 在应用中，多项式的凸性是已知的或期望的。Helton和PI的一个示例结果是，非交换变量的凸多项式的次数至多为2，并且可以写成线性项加上线性项的平方和。麦卡洛将继续扩展非交换多项式和有理函数的理论，并着眼于凸性。 他还将继续调查非自伴算子代数的功能，特别是那些相关的多连接域和几个交换变量的单位球指定由一个给定的收集功能。泛函分析和算子理论的思想和技巧是研究矩阵不等式的有力工具，这些矩阵不等式的形式为多项式或有理函数的矩阵（作为变量）是半正定的。这样的矩阵不等式模型的一大类工程线性系统的问题，如自动控制器的设计和控制中出现的问题。 凸矩阵不等式在设计和数值实现中具有重要意义。 这个提议的一个目标是进一步理解凸矩阵不等式，以及何时以及如何将非凸不等式转换为凸不等式。

The research of this proposal is in functional analysis and operator theory related to engineering system theory and to several variables and mutliply connected domains in the complex plane. Many problems in linear control theory can be formulated in terms of polynomials in several non-commuting variables. In applications, convexity of the polynomial is either known or desired. A sample result of Helton and the PI is that a convex polynomial in non-commuting variables has degree at most two and can be written as a linear term plus a sumof squares of linear terms. McCullough will continue extending the theory of non-commutative polynomials and rational functions with an eye toward convexity. He will also continue to investigate non-self-adjoint operator algebras of functions, particularly those associated to multiply connected domains and to several commuting variables whose unit balls are specified by a given collection of functions. Ideas and techniques from functional analysis and operator theory are powerfultools in the study of matrix inequalities which take the form of a polynomial or rational function of matrices (as the variables) being positive semi-definite. Such matrix inequalities model a large class of engineering linear systems problems, like those arising in the design and control of automatic controllers. Convex matrix inequalities are important for design and numerical implementation. A goal of this proposal is to further understand convex matrix inequalities and when and how it is possible to convert a non-convex inequality into a convex inequality.