Operator functions and the operator harmonic mean
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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
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Additional Information
- Mitsuru Uchiyama
- Affiliation: Department of Mathematics, Shimane University, Matsue Shimane, 690-8504 Japan; and Ritsumeikan University, Kusatsu Siga, 525-8577 Japan
- MR Author ID: 198919
- Email: uchiyama@riko.shimane-u.ac.jp
- 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
- © Copyright 2019 American Mathematical Society
- 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
- MathSciNet review: 4052216