Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Operator functions and the operator harmonic mean


Author: Mitsuru Uchiyama
Journal: Proc. Amer. Math. Soc. 148 (2020), 797-809
MSC (2010): Primary 47A63; Secondary 47A60, 15A39, 26A51
DOI: https://doi.org/10.1090/proc/14753
Published electronically: August 28, 2019
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The objective of this paper is to investigate operator functions by making use of the operator harmonic mean `$ \,!\,$'. For $ 0<A\leqq B$, we construct a unique pair $ X$, $ Y$ such that $ 0<X\leqq Y, \; A=X\,!\,Y,\; B=\frac {X+Y}{2}$. We next give a condition for operators $ A, B, C\geqq 0$ in order that $ C \leqq A\,!\ B$ and show that $ g\ne 0$ is strongly operator convex on $ J$ if and only if $ g(t)>0$ and $ g (\frac {A+B}{2}) \leqq g(A)\,!\,g(B)$ for $ A, B$ with spectra in $ J$. This inequality particularly holds for an operator decreasing function on the right half line. We also show that $ f(t)$ defined on $ (0, b)$ with $ 0<b\leqq \infty $ is operator monotone if and only if $ f(0+)<\infty , \;f (A\,!\,B)\leqq \frac {1}{2}(f(A) + f(B))$. In particular, if $ f>0$, then $ f$ is operator monotone if and only if $ f (A\, !\, B) \leqq f(A)\, !\, f(B)$. We lastly prove that if a strongly operator convex function $ g(t)>0$ on a finite interval $ (a, b)$ is operator decreasing, then $ g$ has an extension $ \tilde {g}$ to $ (a, \infty )$ that is positive and operator decreasing.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 47A63, 47A60, 15A39, 26A51

Retrieve articles in all journals with MSC (2010): 47A63, 47A60, 15A39, 26A51


Additional Information

Mitsuru Uchiyama
Affiliation: Department of Mathematics, Shimane University, Matsue Shimane, 690-8504 Japan; and Ritsumeikan University, Kusatsu Siga, 525-8577 Japan
Email: uchiyama@riko.shimane-u.ac.jp

DOI: https://doi.org/10.1090/proc/14753
Keywords: Operator monotone functions, Loewner theorem, strongly operator convex functions, operator harmonic mean
Received by editor(s): April 13, 2019
Received by editor(s) in revised form: June 26, 2019
Published electronically: August 28, 2019
Additional Notes: The author was supported in part by (JSPS) KAKENHI 17K05286
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2019 American Mathematical Society