Taiwanese Journal of Mathematics

BRIEF SURVEY OF RECENT APPLICATIONS OF AN ORDER PRESERVING OPERATOR INEQUALITY

Takayuki Furuta

Full-text: Open access

Abstract

This short paper surveys recent several applications of an order preserving operator inequality, especially, logarithmic trace inequalities are presented.

Article information

Source
Taiwanese J. Math., Volume 12, Number 8 (2008), 2113-2135.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405138

Digital Object Identifier
doi:10.11650/twjm/1500405138

Mathematical Reviews number (MathSciNet)
MR2459816

Zentralblatt MATH identifier
1179.47017

Subjects
Primary: 47A63: Operator inequalities

Keywords
order preserving operator inequaslity logarithmic trace inequality

Citation

Furuta, Takayuki. BRIEF SURVEY OF RECENT APPLICATIONS OF AN ORDER PRESERVING OPERATOR INEQUALITY. Taiwanese J. Math. 12 (2008), no. 8, 2113--2135. doi:10.11650/twjm/1500405138. https://projecteuclid.org/euclid.twjm/1500405138


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References

  • [4.] T. Ando and F. Hiai, Log majorization and complementary Golden-Thompson type inequalities, Linear Alg. and Its Appl., 197, 198 (1994), 113-131.\hs 0.2cm \hfill (A-3) \item [5.] N. Bebiano, R.Lemos and J. da Providência, Inequalities for quantum relative entropy, Linear Alg. and Its Appl., 401 (2005), 159-172. \hs 0.2cm \hfill(A-3),(A-2) \item [6.] J. I. Fujii and E. Kamei, Relative operator entropy in noncommutative information theory, Math. Japon, 34 (1989), 341-348. \hs 0.2cm \hfill(A-2) \item [7.] M. Fujii, Furuta's inequality and its mean theoretic approach, J. Operator Theory, 232 (1990), 67-72. \hs 0.2cm \hfill($\S1$) \item [8.] M. Fujii, F. Jiang, E. Kamei and K. Tanahashi, A characterization of chaotic order and a problem, J. Inequal. Appl., 2 (1998), 149-156. \hs 0.2cm \hfill(A-1) \item [9.] M.F ujii and E. Kamei, Mean theoretic aspproach to the grand Furuta inequality, Proc. Amer. Math. Soc., 124 (1996), 2751-2756. \hfill ($\S1$) \item [10.] M. Fujii, E. Kamei and R. Nakamoto, Grand Furuta inequality and its variant, J. Math. Inequal., 1 (2007), 437-441. \hfill (A-3) (A-6)($\S1$) \item [11.] 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. \hfill($\S1$) \item [12.] M. Fujii and R. Nakamoto, A geometric mean in the Furuta inequality, Scientiae Mathematicae Japonicae Online, 5 (2001), 435-441. \hfill($\S1$) \item [13.] M. Fujii, R. Nakamoto and M. Tominaga, Generalized Bebiano-Lemos-Providência inequalities and their reverses, Linear Alg. and Its Appl., 426 (2007), 33-39. \hs 0.2cm \hfill(A-3) \item [14.] T. Furuta, $A \ge B \ge 0$ assures $(B^rA^pB^r)^{1/q} \ge B^{(p+2r)/q}$ for $r \ge 0, p \ge 0, q \ge 1$ with \hs 0.3cm $(1+2r)q \ge p+2r$, Proc. Amer. Math. Soc., 101 (1987), 85-88. \hfill($\S1$) \item [15.] T. Furuta, The operator equation $T(H^{\f{1}{n}}T)^n=K$, Linear Alg. Appl., 109 (1988), 149-152. \hfill(C-1) \item [16.] T. Furuta, Elementary proof of an order preserving inequality, Proc. Japan Acad., 65 (1989), 126. \hfill($\S1$) \item [17.] T. Furuta, Applications of order preserving operator inequality, Operator Theory: Advances and Applications, Birkhäuser, 59 (1992), 180-190. \hfill ($\S1$) (A-1) \item [18.] T. Furuta, An extension of the Furuta inequality and Ando-Hiai log majorization, Linear Alg. and Its Appl., 219 (1995), 139-155. \hfill ($\S1$) (A-3), (A-6) \item [19.] T. Furuta, Generalizations of Kosaki trace inequalities and related trace inequalities on chaotic order, Linear Alg. and Its Appl., 235 (1996), 153-161. \hfill(B-3) \item [20.] T. Furuta, Simplified proof of an order preserving operator inequality, Proc. Japan Acad., 74, Ser A, (1998), 114. \hfill ($\S1$), (A-3) \item [21.] T. Furuta, Results under $ \log A \ge \log B$ can be derived from ones under $ A \ge B \ge 0$ by Uchiyama's method -associated with Furuta and Kantorovich type operator inequalities, Math. Inequal. Appl., 3 (2000), 423-436. \hfill(A-7), (A-1) \item [22.] T. Furuta, Invitation to Linear Operators, Taylor & Francis, 2001, London. \hfill($\S1$) (A-1),(A-2), (A-3),(A-4), (A-5), (A-6), (A-7), (B-1), (B-2) \item [23.] T. Furuta, Convergence of logarithmic trace inequalities via generalized Lie-Trotter formulae, Linear Alg. and Its Appl., 396 (2005), 353-372. \hfill (A-3) \item [24.] T. Furuta, Operator inequality implying generalized Bebiano-Lemos-Providência one, Linear Alg. and Its Appl., 426 (2007), 342-348. \hfill(A-3) \item [25.] T. Furuta, Monotonicity of order preserving operator functions, Linear Alg and its Appl., 428 (2008), 1072-1082. \hfill (A-3), (A-6), ($\S1$) \item [26.] T. Furuta, M. Hashimoto and M. Ito, Equivalence relation between generalized Furuta inequality and related operator functions, Scienticae Mathematicae, 1 (1998), 257-259. (A-3), (A-6), ($\S1$) \item [27.] T. Furuta and M. Yanagida, Further extensions of Aluthge transformation on $p$-hyponormal operators, Integral Equations and Operator Theory, 29 (1997), 122-125.\hfill (A-4) \item [28.] T. Furuta, M. Yanagida and T. Yamazaki, Operator functions implyimg Furuta in- equalitty, Math. Inequal. Appl., 1 (1998), 123-130. (A-3), (A-6), ($\S1$) \item [29.] Hiai, Log-majorizations and norm inequalities for exponential operators, Linear operators (Warsaw, 1994), 119-181. Banach Center Publ., 38, Polish Acad. Sci., Warsaw, 1997. \hfill (A-3) \item [30.] Hiai and Petz, The Golden-Thompson trace inequality is complemented, Linear Alg. and Its Appl., 181 (1993), 153-185.\hfill (A-3), (A-6), (A-2) \item [31.] T. Huruya. A note on $p$-hyponormal operators, Proc. Amer. Math. Soc., 125 (1997), 3617-3624. \hfill (A-4) \item [32.] M. Ito. Some classes of operators associated with generalized Aluthge transformation, SUT J. Math., 35 (1999), 149-165.\hfill (A-4) \item [33.] J. F. Jiang, and Kamei and M. Fujii, Operator functions associated with the grand Furuta inequality, Math. Inequal. Appl., 1 (1998), 267-277.\hfill (A-6) \item [34.] E. Kamei, A satellite to Furuta's inequality, Math. Japon, 33 (1988), 883-886. \hfill ($\S1$) \item [35.] E., Kamei, Parametrized, grand, Furuta, inequality, Math. Japon, 50, (1999), 79-83. (A-6), ($\S1$) \item [36.] E. Kamei, Extension of Furuta inequality via generalized Ando-Hiai theorem (Ja- panese), to appear in Surikaisekikenkyūsho Kōkyūroku, Research Institute for Mathematical Sciences, 2007. (A-3), (A-6), ($\S1$) \item [37.] Y. O. Kim, An application of Furuta inequality, Nihonkai Math. J., 10 (1999), 195-198. (A-4), (A-5) \item [38.] H. Kosaki, On some trace inequality, Proc. Centre math. Anal. Austral. Nat. Univ., 1991, pp. 129-134. \hfill (B-3) \item [39.] H. Kosaki, A remark on Sakai's quadratic Radon-Nikodym theorem, Proc. Amer. Math Soc., 116 (1992), 783-786. \hfill ($\S1$) \item [40.] F. Kubo and T. Ando, Means of positive linear operators, Math. Ann., 246 (1980), 205-224. \hfill ($\S1$) \item [41.] C. S. Lin, The Furuta inequality and an operator equation for linear operators, Publ. RIMS, Kyoto Univ., 35 (1999), 309-313.\hfill (C-1), ($\S1$) \item [42.] J. Mićić. and J. Pečarić, Generalization of Kantorovich type operator inequalities via grand Furuta inequality, Math. Inequal. Appl., 9 (2006), 495-510.\hfill (A-7) \item [43.] M. Nakamura and H. Umegaki, A note on entropy for operator algebras, Proc. Japan Acad., 37 (1961), 149-154.\hfill (A-2) \item [44.] G. K. Pedersen and M. Takesaki, The operator equation $THT=K$, Proc. Amer. Math. Soc., 36 (1972), 311-312.\hfill (C-1) \item [45.] T. Sano, Furuta inequality of indefinite type, Math. Inequal. Appl., 10 (2007), 381-387. \hfill ($\S1$) \item [46.] Y. Seo, Kantorovich type operator inequalities for Furuta inequality, Oper. Matrices, 1 (2007), 143-152. \hfill (A-7) \item [47.] K. Tanahashi, Best possibility of the Furuta inequality, Proc. Amer. Math. Soc., 124 (1996), 141-146. \hfill ($\S1$) \item [48.] K. Tanahashi, The best possibility of the grand Furuta inequality, Proc. Amer. Math. Soc., 128 (2000), 511-519.\hfill ($\S1$), (A-3) \item [49.] K. Tanahashi and A. Uchiyama, The Furuta inequality in Banach $*$-algebra, Proc. Amer. Math. Soc., 128 (2000), 1691-1695. \hfill ($\S1$) \item [50.] M. Uchiyama, Some exponential operator inequalities, Math. Inequal. Appl., 2 (1999), 469-471. \hfill(A-1) \item [51.] M. Uchiyama, A new majorization between functions, polynomials, and operator inequalities, J. Func. Anal., 231 (2006), 221-244. \hfill ($\S1$) \item [52.] D. Wang, An operator inequality, Missouri J. Math. Sci., 7 (1995), 17-19. \hfill ($\S1$) \item [53.] T. Yamazaki, Simplified proof of Tanahashi's result on the best possibility of gener- alized Furuta inequality, Math. Inequal. Appl., 2 (1999), 473-477. ($\S1$), (A-3) \item [54.] T. Yamazaki, An expression of spectral radius via Aluthge transformation, Proc. Amer. Math. Soc., 130 (2002), 1131-1137. \hfill (A-4) \item [55.] M. Yanagida, Some applications of Tanahashi's result on the best possibility of gen- eralized Furuta inequality, Math. Inequal. Appl., 2 (1999), 297-305. ($\S1$), (A-3) \item [56.] M. Yanagida, Powers of class $wA(s,t)$ operators associated with generalized Aluthge transformation, J. Inequal. Appl., 7 (2002), 143-168. \hfill (A-4) \item [57.] T. Yoshino, The $p$-hyponormality of the Aluthge transformation, Interdiscip. Inform. Sci., 3 (1997), 91-93. \hfill (A-4) \item [58.] J. Yuan and Z. Gao, The Furuta inequality and Furuta type operator functions under chaotic order, Acta Sci. Math., $($Szeged$)$, 73 (2007), 669-681. \hfill (A-6) \item [59.] J. Yuan and Z. Gao, Complete form of Furuta inequality, to appear in Proc. Amer. Math. Soc. \hfill (A-6), ($\S1$) \item [60.] H. Umegaki, Conditional expectation in an operator algebra IV, Kodai Math. Sem. Rep., 14 (1962), 59-85. \hfill(A-2)