Operator Theory and Stable Polynomials

Washington University, Saint Louis MO

Investigators

Abstract

Operator theory is a broad and mature area of pure mathematics with close ties to the mathematics of quantum mechanics and control systems engineering. Indeed, von Neumann is often credited with formulating the foundations of operator theory as a language for quantum mechanics, while Nobert Wiener initiated an approach to engineering prediction problems (such as jet tracking) based on ideas from operator theory and harmonic analysis. In more recent decades, operator theory has been used in H-infinity control theory which has applications in automatic pilot design. On the other hand, a stable polynomial is not a field per se but a basic concept in mathematics that has become profoundly useful in the study of a diverse range of problems in mathematics: combinatorics (enumeration problems), graph theory (the study of networks), and operator theory. The purpose of this project is to use operator theory to study stable polynomials and vice versa. This project focuses on three problems: (1) the generalized Lax conjecture, (2) a quantitative understanding of linear preservers of stability, and (3) extensions of the theory of stable polynomials to entire functions. Semi-definite programming is a powerful technique in optimization and the generalized Lax conjecture is a bold assertion about the sets on which this theory can successfully be implemented. The conjecture claims that these feasibility sets have a geometric description in terms of hyperbolic polynomials (a slight generalization of the concept of a stable polynomial). Although evidently important in optimization, this would further establish links between stable polynomials and operator theory as the question is closely related to the concept of representing stable polynomials via operator theoretic determinantal representations. Problem (2) is about taking the highly successful theorems of Borcea-Branden that characterized the linear maps on stable polynomials that preserve stability and making them quantitative. How exactly do stability preservers modify zero sets? One approach to this problem is through analyzing how stability preservers affect various sums-of-squares formulas for stable polynomials. Problem (3) is about a natural extension of stable polynomials to various classes of entire functions (the Polya class, Hermite-Biehler class) and would require developing multivariable theories of Hilbert spaces of entire functions. This latter endeavor is important in its own right as the successes in one variable Hilbert spaces of entire functions have been profound (i.e. the Poltoratski approach to problems of uncertainty type). 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.