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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Uwe Franz, Fumio Hiai and Éric Ricard
Journal: Trans. Amer. Math. Soc. 366 (2014), 3043-3074
MSC (2010): Primary 47A56, 47A60, 47A63, 15A39
Published electronically: February 17, 2014
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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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Additional Information

Uwe Franz
Affiliation: Département de mathématiques de Besançon, Université de Franche-Comté, 16, route de Gray, 25 030 Besançon Cedex, France
Email: uwe.franz@univ-fcomte.fr

Fumio Hiai
Affiliation: Tohoku University (Emeritus), Hakusan 3-8-16-303, Abiko 270-1154, Japan
Email: hiai.fumio@gmail.com

Éric Ricard
Affiliation: Laboratoire de Mathématiques Nicolas Oresme, Université de Caen Basse-Normandie, BP 5186, 14032 Caen Cedex, France
Email: eric.ricard@unicaen.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-05942-4
PII: S 0002-9947(2014)05942-4
Keywords: Operator monotone function, operator convex function, matrix monotone function, matrix convex function, divided difference, operator $k$-tone function, matrix $k$-tone function, absolutely monotone function, completely monotone function.
Received by editor(s): June 7, 2011
Received by editor(s) in revised form: August 3, 2012
Published electronically: February 17, 2014
Article copyright: © Copyright 2014 American Mathematical Society