An operator inequality related to Jensen’s inequality
HTML articles powered by AMS MathViewer
- by Mitsuru Uchiyama PDF
- Proc. Amer. Math. Soc. 129 (2001), 3339-3344 Request permission
Abstract:
For bounded non-negative operators $A$ and $B$, Furuta showed \[ 0\leq A \leq B \ \textrm {implies } \ A^{\frac {r}{2}}B^sA^{\frac {r}{2}} \leq (A^{\frac {r}{2}}B^t A^{\frac {r}{2}})^{\frac {s+r}{t+r}} \ \ (0\leq r, \ 0\leq s \leq t).\] We will extend this as follows: $0\leq A\leq B \underset {\lambda }{!}C$ $(0<\lambda <1)$ implies \[ A^{\frac {r}{2}}(\lambda B^s+ (1-\lambda )C^s)A^{\frac {r}{2}} \leq \{A^{\frac {r}{2}} (\lambda B^t+ (1- \lambda )C^t) A^{\frac {r}{2}}\}^{\frac {s+r}{t+r}} ,\] where $B \underset {\lambda }{!}C$ is a harmonic mean of $B$ and $C$. The idea of the proof comes from Jensen’s inequality for an operator convex function by Hansen-Pedersen.References
- W. N. Anderson, G. E. Trapp, Shorted operators II, SIAM J. Appl. Math. 28(1975), 60–71.
- W. N. Anderson Jr. and G. E. Trapp, Shorted operators. II, SIAM J. Appl. Math. 28 (1975), 60–71. MR 356949, DOI 10.1137/0128007
- Masatoshi Fujii, Takayuki Furuta, and Eizaburo Kamei, Furuta’s inequality and its application to Ando’s theorem, Linear Algebra Appl. 179 (1993), 161–169. MR 1200149, DOI 10.1016/0024-3795(93)90327-K
- Takayuki Furuta, $A\geq B\geq 0$ assures $(B^rA^pB^r)^{1/q}\geq B^{(p+2r)/q}$ for $r\geq 0$, $p\geq 0$, $q\geq 1$ with $(1+2r)q\geq p+2r$, Proc. Amer. Math. Soc. 101 (1987), no. 1, 85–88. MR 897075, DOI 10.1090/S0002-9939-1987-0897075-6
- Takayuki Furuta, Two operator functions with monotone property, Proc. Amer. Math. Soc. 111 (1991), no. 2, 511–516. MR 1045135, DOI 10.1090/S0002-9939-1991-1045135-2
- Frank Hansen and Gert Kjaergȧrd Pedersen, Jensen’s inequality for operators and Löwner’s theorem, Math. Ann. 258 (1981/82), no. 3, 229–241. MR 649196, DOI 10.1007/BF01450679
- Fumio Kubo and Tsuyoshi Ando, Means of positive linear operators, Math. Ann. 246 (1979/80), no. 3, 205–224. MR 563399, DOI 10.1007/BF01371042
- Kôtarô Tanahashi, Best possibility of the Furuta inequality, Proc. Amer. Math. Soc. 124 (1996), no. 1, 141–146. MR 1291794, DOI 10.1090/S0002-9939-96-03055-9
- Mitsuru Uchiyama, Some exponential operator inequalities, Math. Inequal. Appl. 2 (1999), no. 3, 469–471. MR 1698384, DOI 10.7153/mia-02-36
- M. Uchiyama, Strong monotonicity of operator functions, Integr. Equ. Oper. Theory 37(1) (2000), 95–105.
Additional Information
- Mitsuru Uchiyama
- Affiliation: Department of Mathematics, Fukuoka University of Education, Munakata, Fukuoka, 811-4192, Japan
- MR Author ID: 198919
- Email: uchiyama@fukuoka-edu.ac.jp
- Received by editor(s): March 21, 2000
- Published electronically: April 9, 2001
- Communicated by: Joseph A. Ball
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 3339-3344
- MSC (2000): Primary 47A63, 15A48
- DOI: https://doi.org/10.1090/S0002-9939-01-06130-5
- MathSciNet review: 1845011