Annals of Functional Analysis

Furuta Inequality and its related topics

Masatoshi Fujii

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‎This article is devoted to a brief survey of Furuta inequality and its related topics‎. ‎It consists of 4 sections‎: ‎1‎. ‎From L\"owner-Heinz inequality to Furuta inequality‎, ‎2‎. ‎Ando--Hiai inequality‎, ‎3‎. ‎Grand Furuta inequality‎, ‎and 4‎. ‎Chaotic order‎.

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Ann. Funct. Anal., Volume 1, Number 2 (2010), 24-45.

First available in Project Euclid: 12 May 2014

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Zentralblatt MATH identifier

Primary: 47A63: Operator inequalities
Secondary: 47A64: Operator means, shorted operators, etc.

L\"owner-Heinz inequality ‎Furuta inequality ‎Ando--Hiai inequality ‎Grand Furuta inequality ‎chaotic order and operator geometric mean


Fujii, Masatoshi. Furuta Inequality and its related topics. Ann. Funct. Anal. 1 (2010), no. 2, 24--45. doi:10.15352/afa/1399900585.

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  • T. Ando, Topics on Operator Inequalities, Lecture notes (mimeographed), Hokkaido Univ., Sapporo, 1978.
  • T. Ando, On some operator inequality, Math. Ann. 279 (1987), 157–159.
  • T. Ando and F. Hiai, Log-majorization and complementary Golden-Thompson type inequalities, Linear Algebra Appl. 197,198 (1994), 113–131
  • N.N. Chan and M.K. Kwong, Hermitian matrix inequalities and a conjecture, Amer. Math. Monthly 92 (1985), 533–541.
  • M. Fujii, Furuta's inequality and its mean theoretic approach, J. Operator Theorey 23 (1990), 67–72.
  • M. Fujii, T. Furuta and E. Kamei, Furuta's inequality and its application to Ando's theorem, Linear Algebra Appl. 179 (1993),161–169.
  • M. Fujii, M. Hashimoto, Y. Seo and M. Yanagida, Characterizations of usual and chaotic order via Furuta and Kantorovich inequalities, Sci. Math. 3 (2000), 405–418.
  • M. Fujii, J.-F. Jiang and E. Kamei, Characterization of chaotic order and its application to Furuta inequality, Proc. Amer. Math. Soc. 125 (1997), 3655–3658.
  • M. Fujii, J.-F. Jiang, E. Kamei and K. Tanahashi, A characterization of chaotic order and a problem, J.Inequal. Appl. 2 (1998), 149–156.
  • M. Fujii and E. Kamei, Mean theoretic approach to the grand Furuta inequality, Proc. Amer. Math. Soc. 124 (1996), 2751–2756.
  • M. Fujii and E. Kamei, Ando–Hiai inequality and Furuta inequality, Linear Algebra Appl. 416 (2006), 541–545.
  • M. Fujii and E. Kamei, Variants of Ando–Hiai inequality, Operator Theory: Adv. Appl. 187 (2008), 169–174.
  • M. Fujii, A. Matsumoto and R. Nakamoto, A short proof of the best possibility for the grand Furuta inequality, J. Inequal. Appl. 4 (1999), 339–344.
  • T. 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), 85–88.
  • T. Furuta, Elementary proof of an order preserving inequality, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), 126.
  • T. Furuta, Applications of order preserving operator inequalities, Operator theory and complex analysis (Sapporo, 1991), 180–190, Oper. Theory Adv. Appl., 59, Birkhüser, Basel, 1992.
  • T. Furuta, Extension of the Furuta inequality and Ando–Hiai log-majorization, Linear Algebra Appl. 219 (1995), 139–155.
  • T. Furuta, Simplified proof of an order preserving operator inequality, Proc. Japan. Acad. 74 (1998), 114.
  • T. Furuta, Invitation to Linear Operators, Taylor&Francis, London, 2001.
  • E. Heinz, Beitrage zur Storungstheorie der Spectralzegung, Math. Ann. 123 (1951), 415–438.
  • E. Kamei, A satellite to Furuta's inequality, Math. Japon. 33 (1988), 883–886.
  • F. Kubo and T. Ando, Means of positive linear operators, Math. Ann. 246 (1980), 205–224.
  • K. Löwner, Über monotone Matrix function, Math. Z. 38 (1934), 177–216.
  • G.K. Pedersen, Some operator monotone functions, Proc. Amer. Math. Soc. 36(1972), 309–310.
  • K. Tanahashi, Best possibility of the Furuta inequality, Proc. Amer. Math. Soc. 124 (1996), 141–146.
  • K. Tanahashi, The best possibility of the grand Furuta inequality, Proc. Amer. Math. Soc. 128 (2000), 511–519.
  • M. Uchiyama, Some exponential operator inequalities, Math. Inequal. Appl. 2 (1999), 469–471.
  • T. Yamazaki, Simplified proof of Tanahashi's result on the best possibility of generalized Furuta inequality, Math. Inequal. Appl. 2 (1999), 473–477.
  • M. Yanagida, Some applications of Tanahashi's result on the best possibility of Furuta inequality, Math. Inequl. Appl. 2 (1999), 297–305.