Higher order extension of Löwner’s theory: Operator 𝑘-tone functions

Abstract

Abstract. The new notion of operator/matrix k-tone functions is introduced, which is a higher order extension of operator/matrix monotone and convex functions. Differential properties of matrix k-tone functions are shown. Characterizations, properties, and examples of operator k-tone functions are presented. In particular, integral representations of operator k-tone functions are given, generalizing familiar representations of operator monotone and convex functions.

Conic structure of the non-negative operator convex functions on $(0,\infty)$

“…As seen in the proof of [5, Theorem 1.10], the measure ν (hence µ) and γ ≥ 0 are unique in this expression of h. The discussion is similar for f in Theorem 2.1 (see the proof of [5,Theorem 1.8]). We call the measure µ in the above theorems the representing measure of f .…”

“…The notion of operator k-tone functions in [5] extends operator monotone and operator convex functions. In fact, operator 1-tone and operator 2-tone functions are operator monotone and operator convex functions, respectively.…”

“…In the previous paper [5] we introduced the notion of operator k-tone functions on an open interval, extending operator monotone and convex functions. The integral representations for operator k-tone functions are useful here, so that we state them in Section 2 of this paper in the form specialized to the case k = 2 (the case of operator convex functions).…”

Families of completely positive maps associated with monotone metrics

“…The integral expression (A.1) was also given in [33], which was considerably extended in [13,Theorem 5.1]. There is another route to prove the two theorems in Section 2. when k(x) is given by (4.4).…”

“…Since the divided difference function h(x) ≡ (k(x) − k(1))/(x − 1) is operator monotone on (0, ∞) as in Section A.1, it is known (see, e.g., [13 1 + x (x + λ)(1 + λx) · (1 + λ) 2 2 dm(λ).…”