Operator Theory and Matrix Inequalities

University Of Florida, Gainesville FL

Investigators

Abstract

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.