Manipulating matrix inequalities

Abstract: A natural partial ordering on self-adjoint matrices is to set A <= B whenever B – A is positive semidefinite. We say a function f:(a,b) -> R is matrix monotone in the sense that A <= B implies f(A) <= f(B), where f is evaluated using the functional calculus. (If the function f were a rational function, f(A) is simply defined by substituting A into f.) In 1934, Charles Loewner showed that a function is matrix monotone if and only if it is analytic and analytically continues to the upper half plane in C as a map into the closed upper half plane. With Ryan Tully-Doyle, we characterized the appropriate generalization for systems of matrix inequalities by proving a non-commutative version of Loewner’s theorem. Our work unites some of the basic theory already encountered in the wild in the study of matrix means, the Schur complement, random matrix theory, complete positivity, free probability and free analysis. In this talk, we will give the classical theory and some of its applications, then proceed to describe the notion of non-commutative function theory, and finally describe my current work on the non-commutative Loewner theorem with a several examples and applications.