Characterization of chaotic order

and its application to furuta inequality

Authors:
Masatoshi Fujii, Jian Fei Jiang and Eizaburo Kamei

Journal:
Proc. Amer. Math. Soc. **125** (1997), 3655-3658

MSC (1991):
Primary 47A63, 47B15

MathSciNet review:
1415586

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Abstract | References | Similar Articles | Additional Information

Abstract: In this note, we give a simple characterization of the chaotic order among positive invertible operators on a Hilbert space. As an application, we discuss Furuta's type operator inequality.

**1.**T. Ando,*On some operator inequalities*, Math. Ann.**279**(1987), no. 1, 157–159. MR**912844**, 10.1007/BF01456197**2.**Masatoshi Fujii,*Furuta’s inequality and its mean theoretic approach*, J. Operator Theory**23**(1990), no. 1, 67–72. MR**1054816****3.**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**, 10.1016/0024-3795(93)90327-K**4.**Masatoshi Fujii and Eizaburo Kamei,*Furuta’s inequality and a generalization of Ando’s theorem*, Proc. Amer. Math. Soc.**115**(1992), no. 2, 409–413. MR**1091180**, 10.1090/S0002-9939-1992-1091180-1**5.**Takayuki Furuta,*𝐴≥𝐵≥0 assures (𝐵^{𝑟}𝐴^{𝑝}𝐵^{𝑟})^{1/𝑞}≥𝐵^{(𝑝+2𝑟)/𝑞} for 𝑟≥0, 𝑝≥0, 𝑞≥1 with (1+2𝑟)𝑞≥𝑝+2𝑟*, Proc. Amer. Math. Soc.**101**(1987), no. 1, 85–88. MR**897075**, 10.1090/S0002-9939-1987-0897075-6**6.**Takayuki Furuta,*An elementary proof of an order preserving inequality*, Proc. Japan Acad. Ser. A Math. Sci.**65**(1989), no. 5, 126. MR**1011850****7.**Eizaburo Kamei,*A satellite to Furuta’s inequality*, Math. Japon.**33**(1988), no. 6, 883–886. MR**975867****8.**Gert K. Pedersen,*Some operator monotone functions*, Proc. Amer. Math. Soc.**36**(1972), 309–310. MR**0306957**, 10.1090/S0002-9939-1972-0306957-4

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Additional Information

**Masatoshi Fujii**

Affiliation:
Department of Mathematics, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka 582, Japan

Email:
mfujii@cc.osaka-kyoiku.ac.jp

**Jian Fei Jiang**

Affiliation:
Department of Mathematics, Osaka Kyoiku University, Kashiwara, Osaka 582, Japan;
permanent address: Department of Basic Science and Technology, China Textile University, Shanghai, China, Postal code 200051

**Eizaburo Kamei**

Affiliation:
Momodani Senior High School, Ikuno, Osaka 544, Japan

Address at time of publication:
Maebashi Institute of Technology, Kamisadori, Maebashi, Gunma 371, Japan

Email:
kamei@maebashi-it.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-97-04057-4

Keywords:
Positive operators,
chaotic order,
L\"{o}wner-Heinz inequality,
Furuta inequality

Received by editor(s):
July 16, 1996

Dedicated:
Dedicated to Professor P. R. Halmos on his 80th Birthday

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1997
American Mathematical Society