Operator Theory and Matrix Inequalities

许多实际问题和数学问题都可以用多项式不等式来描述。这些是代数不等式，它们的变量(未知数)代表数字。由于多项式的重要性，多项式不等式已经被深入研究了许多世纪。矩阵不等式，即以矩阵为变量的代数方程，也出现在许多应用领域，如线性系统工程和数学领域。多项式和矩阵不等式之间的一个关键区别是，对于矩阵X和Y，与实数不同，XY和YX可能不相同，如果X和Y是“正的”，则XY不一定是“正的”。这个项目的一个主要主题是发展一种强调凸性的矩阵不等式的数学理论。如果连接集合中任意两点的线段完全位于集合中，则集合是凸的。虽然很容易画出两个变量的凸集图，但这个概念在有大量变量的情况下是有意义的，并且在有大量变量的情况下非常重要，例如最小化成本或在设计问题中确定最差情况的性能是至关重要的。在许多情况下，当在凸集上搜索时，存在唯一的最小值(或最坏情况)，并且有非常可靠的计算机算法来寻找最小值。另一方面，当在非凸集上搜索最小值时，很难确定找到了最小值，而不是简单地小于附近那些值的值。线性矩阵不等式的解集是凸的，但一般情况下矩阵不等式的解集不是凸的。主要研究人员和他的合作者的一个目标是确定矩阵不等式的解集何时是凸的，何时可以以一种容易处理的方式转换为用线性矩阵不等式描述的凸集，或者可以合理地近似为凸集。这个项目将有助于凸矩阵不等式的数学基础，将这门学科平行于半代数几何，并发展为算子系统和空间理论以及完全正/压缩映射的非线性版本。动力来自于与工程系统理论等应用程序的深度科学互动。它将促进对线性矩阵不等式的理解，线性矩阵不等式现在是科学和工程中的标准工具，以及系统工程问题中出现的矩阵不等式，特别是对于许多从信号流程图中出现的问题。一个重要的目标是确定矩阵不等式与凸解集的关系并使其自动化，将非凸解集映射到凸解集(目前在系统工程文献中的某些特殊情况下是在特别的基础上完成的)，否则用凸解集逼近非凸解集。所采用的方法涉及泛函分析、算子理论、复变数、算子系统和空间、完全正映射、半定规划和半代数几何的思想。相反，这一建议的一个重要方面是技术和结果的发展，这些技术和结果有助于算子系统、空间、代数和完全正映射的理论，以及一元和多变量解析函数理论的(自由)非交换类似。在另一个方向，将研究自然算子代数上出现在一个和几个复变量中的函数的正映射和完全正映射之间的区别。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Many practical and mathematical problems can be described by polynomial inequalities. These are algebraic inequalities whose variables (unknowns) represent numbers. Because their importance, polynomial inequalities have been studied intensely for many centuries. Matrix inequalities, algebraic equations whose variables are matrices, also appear in many applied areas, such as linear systems engineering, and mathematical fields. A key difference between polynomial and matrix inequalities is that for matrices X and Y, unlike real numbers, it can happen that XY and YX are not the same and if X and Y are "positive", XY need not be "positive". A major theme of this project is the development of a mathematical theory of matrix inequalities with an emphasis on convexity. A set is convex if the line segment joining any two points in the set lies entirely in the set. While it is easy to draw pictures of convex sets in two variables, the notion makes sense and is immensely important in the cases where there are a large number of variables, for instance minimizing cost or in design problems where it is crucial to identifying worst case performance. In many settings, when searching over a convex set, there is a unique minimum (or worst case) and there are very reliable computer algorithms to find the minimum. On the other hand, when searching for a minimum over a set that is not convex it is difficult to be certain that the minimum, and not a value that is simply smaller than those nearby, has been found. The solution set of a linear matrix inequality is convex, but in general solution sets to matrix inequalities are not. A goal of the principal investigator and his collaborators is to determine when the solution set of a matrix inequality is convex, can be converted in a tractable way to, or approximated reasonably by, a convex set described in some fashion by a linear matrix inequality.This project will contribute to the mathematical foundations of convex matrix inequalities, developing the subject both in parallel to semialgebraic geometry and as a non-linear version of the theory of operator systems and spaces and completely positive/contractive maps. Motivation flows from the deep scientific interactions with applications such as engineering systems theory. It will advances the understanding of both linear matrix inequalities, now standard tools in science and engineering, and matrix inequalities that arise in systems engineering problems, particular for many of those that arise from a signal flow diagram. An important aim is to determine and automate the identification of matrix inequalities with convex solution sets, mapping a non-convex solution set to a convex one (currently done on an ad-hoc basis for some special cases in the systems engineering literature), and otherwise approximating non-convex solution sets by convex ones. The methods employed involve ideas from functional analysis, operator theory, complex analysis and several complex variables, operator systems and spaces, completely positive maps, semidefinite programming and semialgebraic geometry. Conversely, a significant aspect of this proposal is the development of techniques and results that contribute to the theories of operator systems, spaces and algebras and completely positive maps and of (freely) noncommutative analogs of the theory of analytic functions in one and several variables. In another direction, the distinction between positive and completely positive maps on natural operator algebras of functions arising in one and several complex variables will be investigated.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.