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Open AccessArticle

Characterizations of Positive Operator-Monotone Functions and Monotone Riemannian Metrics via Borel Measures

1
Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand
2
Department of Biology, Faculty of Biological Sciences, Tokai University, 5-1-1-1 Minamisawa, Minamiku, Sapporo 005-8601, Japan
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(12), 1162; https://doi.org/10.3390/math7121162
Received: 18 November 2019 / Revised: 27 November 2019 / Accepted: 27 November 2019 / Published: 2 December 2019
(This article belongs to the Special Issue Differential Geometry: Theory and Applications)
We show that there is a one-to-one correspondence between positive operator-monotone functions on the positive reals, monotone Riemannian metrics, and finite positive Borel measures on the unit interval. This correspondence appears as an integral representation of weighted harmonic means with respect to that measure on the unit interval. We also investigate the normalized/symmetric conditions for operator-monotone functions. These conditions turn out to characterize monotone metrics and Morozowa–Chentsov functions as well. Concrete integral representations of such functions related to well-known monotone metrics are also provided. Moreover, we use this integral representation to decompose positive operator-monotone functions. Such decomposition gives rise to a decomposition of the associated monotone metric. View Full-Text
Keywords: operator-monotone function; monotone Riemannian metric; Morozowa–Chentsov function; Borel measure; density matrix operator-monotone function; monotone Riemannian metric; Morozowa–Chentsov function; Borel measure; density matrix
MDPI and ACS Style

Chansangiam, P.; Sabau, S.V. Characterizations of Positive Operator-Monotone Functions and Monotone Riemannian Metrics via Borel Measures. Mathematics 2019, 7, 1162.

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