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

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.

final version, 33 pages

References in corpus (4)

• Operator log-convex functions and operator means
• Trace functions as Laplace transforms
• Differential analysis of matrix convex functions II
• Simplicial Differential Calculus, Divided Differences, and Construction of Weil Functors

Cited by in corpus (6)

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• Quantum $f$-divergences in von Neumann algebras II. Maximal $f$-divergences
• Contraction coefficients for noisy quantum channels
• Families of completely positive maps associated with monotone metrics
• Ando-Hiai type inequalities for operator means and operator perspectives