Monotonicity in several non-commuting variables

Let f : (a, b)R. The function f is said to be matrix monotone if A ≤ B implies f(A) ≤ f(B) for all pairs of likesized self-adjoint matrices with spectrum in (a, b). Classically, Charles Loewner showed that a bounded Borel function is matrix monotone if and only if it is analytic and extends to be a self-map of the upper half plane. The theory of matrix montonicity has profound consequences for any general theory of matrix inequalities. For example, it might seem surprising that X ≤ Y does not imply that X ≤ Y , which is a consequence of Loewner’s theorem. We will discuss commutative and noncommutative generalizations to several variables of Loewner’s theorem