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

Pattrawut Chansangiam and Sorin V. Sabau
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Pattrawut Chansangiam: Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand
Sorin V. Sabau: Department of Biology, Faculty of Biological Sciences, Tokai University, 5-1-1-1 Minamisawa, Minamiku, Sapporo 005-8601, Japan

Mathematics, 2019, vol. 7, issue 12, 1-14

Abstract: 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.

Keywords: operator-monotone function; monotone Riemannian metric; Morozowa–Chentsov function; Borel measure; density matrix (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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