On monotone matrix functions of two variables

Abstract:

The theory of monotone matrix functions has been developed by K. Loewner; he first gives some necessary and sufficient conditions for a function to be a monotone matrix function of order n, and then, as a result of further deep investigations including questions of interpolation he arrives at the following criterion: A real-valued function $f(x)$ defined in (a, b) is monotone of arbitrary high order n if and only if it is analytic in (a, b), can be analytically continued onto the entire upper half-plane, and has there a nonnegative imaginary part. The problem of monotone operator functions of two real variables has recently been considered by A. Koranyi. He has generalized Loewner’s theorem on monotone matrix functions of arbitrary high order n to two variables. We seek a theory of monotone matrix functions of two variables analogous to that developed by Loewner and show that a complete analogue to Loewner’s theory exists in two dimensions.