Operator monotone functions, positive definite kernels and majorization

Abstract

Abstract. Let f (t) be a real continuous function on an interval, and consider the operator function f (X) defined for Hermitian operators X. We will show that if f (X) is increasing w.r.t. the operator order, then for F (t) = f (t)dt the operator function F (X) is convex. Let h(t) and g(t) be C 1 functions defined on an interval I. Suppose h(t) is non-decreasing and g(t) is increasing. Then we will define the continuous kernel function K h, g by K h, g (t, s) = (h(t) − h(s))/(g(t) − g(s)), which is a generalization of the Löwner kernel function. We will see that it is positive definite if and only if h(A) h(B) whenever g(A) g(B)for Hermitian operators A, B, and we give a method to construct a large number of infinitely divisible kernel functions.

Matrix Analysis: Matrix Monotone Functions, Matrix Means, and Majorization

“…The next theorem shows Bhatia and Sano's characterizations in [20] for operator convex functions on ð0; 1Þ with some improvements. A similar improvement was also obtained in [76] by a different method. (i) f is operator convex;…”

From positive to accretive matrices

“…Operator monotone functions and operator concave functions are strongly related, as follows [27,Theorem 2.4] and [4,Theorems 2.1,2.3,3.1,3.7].…”

A Note on Kantorovich and Ando Inequalities

“…Unlike scalar monotony and convexity, operator monotony and convexity are strongly related, as stated in the next proposition, which can be found in [19,Theorem 2.4] and [1, Theorem 2.1, Theorem 3.1, Theorem 2.3 and Theorem 3.7].…”