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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Operator functions and the operator harmonic mean
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by Mitsuru Uchiyama PDF
Proc. Amer. Math. Soc. 148 (2020), 797-809 Request permission

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.
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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