Theory of Functions of Noncommuting Variables and Its Applications

Polynomials, rational functions, and formal power series in (free) noncommuting variables were considered in a variety of settings. While usually viewed as formal algebraic objects, they also appeared often as functions by substituting tuples of matrices or operators for the variables. Our point of view is that a function of noncommuting variables is a function defined on tuples of matrices of all sizes that satisfies certain compatibility conditions as we vary the size of matrices: it respects direct sums and simultaneous similarities, or equivalently, simultaneous intertwinings. This leads naturally to a noncommutative difference-differential calculus. The objective of our research is to develop a comprehensive theory of noncommutative functions and their difference-differential calculus in both algebraic and analytic setting. We expect this theory to have a wide range of applications from noncommutative spectral theory (compare Taylor [8, 9]) and free probability (compare Voiculsecu [10, 11]) to analysis of linear matrix inequalities (LMIs) in optimization and control (compare Helton [1], Helton–McCullough–Vinnikov [2], Helton–McCullough–Putinar–Vinnikov [3]).