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A Survey on Operator Monotonicity, Operator Convexity, and Operator Means

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  • Pattrawut Chansangiam

Abstract

This paper is an expository devoted to an important class of real-valued functions introduced by Löwner, namely, operator monotone functions. This concept is closely related to operator convex/concave functions. Various characterizations for such functions are given from the viewpoint of differential analysis in terms of matrix of divided differences. From the viewpoint of operator inequalities, various characterizations and the relationship between operator monotonicity and operator convexity are given by Hansen and Pedersen. In the viewpoint of measure theory, operator monotone functions on the nonnegative reals admit meaningful integral representations with respect to Borel measures on the unit interval. Furthermore, Kubo-Ando theory asserts the correspondence between operator monotone functions and operator means.

Suggested Citation

  • Pattrawut Chansangiam, 2015. "A Survey on Operator Monotonicity, Operator Convexity, and Operator Means," International Journal of Analysis, Hindawi, vol. 2015, pages 1-8, November.
    • Handle: RePEc:hin:ijanal:649839
    DOI: 10.1155/2015/649839
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    1. Pattrawut Chansangiam, 2015. "Positivity, Betweenness, and Strictness of Operator Means," Abstract and Applied Analysis, Hindawi, vol. 2015, pages 1-5, May.
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    Cited by:

    1. Tingting Cai & Dongmin Yu & Huanan Liu & Fengkai Gao, 2022. "RETRACTED: Computational Analysis of Variational Inequalities Using Mean Extra-Gradient Approach," Mathematics, MDPI, vol. 10(13), pages 1-14, July.

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