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

Franz, Uwe; Hiai, Fumio; Ricard, Éric

doi:10.1090/S0002-9947-2014-05942-4

Higher order extension of Löwner’s theory: Operator $k$-tone functions
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by Uwe Franz, Fumio Hiai and Éric Ricard PDF

Trans. Amer. Math. Soc. 366 (2014), 3043-3074 Request permission

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.

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