Matrix power means and new characterizations of operator monotone functions

Abstract

For positive definite matrices A and B, the Kubo-Ando matrix power mean is defined as $P_{\mu }(p,A,B)=A^{1/2}{\left(\frac{1+(A^{-1/2}B{A}^{-1/2}{)}^p}{2}\right)}^{1/p}{A}^{1/2}(p\ge 0).$ In this paper, for $0\le p\le 1\le q$, we show that if one of the following inequalities $f(P_{\mu }(p,A,B))\le f(P_{\mu }(1,A,B))\le f(P_{\mu }(q,A,B))$ holds for any positive definite matrices A and B, then the function f is operator monotone on $(0,\mathrm{\infty }).$ We also study the inverse problem for non-Kubo-Ando matrix power means with the powers 1/2 and 2. As a consequence, we establish new charaterizations of operator monotone functions with the non-Kubo-Ando matrix power means.

Keywords:

• Matrix power mean
• matrix geometric mean
• operator monotone functions
• inverse problems for means
• Kubo-Ando means

2010 Mathematics Subject Classifications:

• 47A63
• 47A64
• 47A56
• 46E05
• 15B48

Acknowledgments

The authors are very grateful to Prof. Mikael de la Salle for useful discussions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The Van Nguyen was funded by Vingroup Joint Stock Company and supported by the Domestic Master Scholarship Programme of Vingroup Innovation Foundation (VINIF), Vingroup Big Data Institute (VINBIGDATA), code VINIF.2020.ThS.KHTN.05 Trung Hoa Dinh was funded by a research grant of Troy University (2021/2022).