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The best possibility
of the grand Furuta inequality


Author: Kôtarô Tanahashi
Journal: Proc. Amer. Math. Soc. 128 (2000), 511-519
MSC (1991): Primary 47B15
DOI: https://doi.org/10.1090/S0002-9939-99-05261-2
Published electronically: July 6, 1999
MathSciNet review: 1654088
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ A, B \in B(H)$ be invertible bounded linear operators on a Hilbert space $H$ satisfying $O\leq B \leq A$ , and let $ p, r, s , t $ be real numbers satisfying $ 1 < s, 0 < t < 1 , t \leq r , 1 \leq p . $ Furuta showed that if $ 0 < \alpha \leq \dfrac{ 1-t+r}{ (p-t)s + r} $, then $ \left\{ A^{\frac{r}{2}} \left( A^{ -\frac{t}{2}} B^{p} A^{ -\frac{t}{2}} \right)^{s} A^{\frac{r}{2}} \right\}^{\alpha } \leq A^{ \left\{ (p-t)s + r \right\} \alpha } $. This inequality is called the grand Furuta inequality, which interpolates the Furuta inequality $(t=0)$
and the Ando-Hiai inequality ( $ t=1, r = s $ ).

In this paper, we show the grand Furuta inequality is best possible in the following sense: that is, if $ \dfrac{ 1-t+r}{ (p-t)s + r} < \alpha $, then there exist invertible matrices $A,B$ with $O\leq B \leq A$ which do not satisfy $ \left\{ A^{\frac{r}{2}} \left( A^{ -\frac{t}{2}} B^{p} A^{ -\frac{t}{2}} \right)^{s} A^{\frac{r}{2}} \right\}^{\alpha } \leq A^{ \left\{ (p-t)s + r \right\} \alpha } $.


References [Enhancements On Off] (What's this?)

  • [1] N. N. Chan and M. K. Kwong, Hermitian matrix inequalities and a conjecture, Amer. Math. Monthly, 92 (1985), 533-541. MR 87d:15011
  • [2] A. Aluthge, Some generalized theorems on $p$-hyponormal operators, Integr. Equat. Oper. Th., 24 (1994), 497-501. MR 97a:47032
  • [3] M. Fujii, T. Furuta, E. Kamei Complements to the Furuta inequality, Journal of the Japan Academy, 70 (1994), 239-242. MR 95j:47018
  • [4] T. Furuta, $A \geq B \geq O$ assures $(B^{r}A^{p}B^{r})^{\frac{1}{q}} \geq B^{\frac{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. MR 89b:47028
  • [5] T. Furuta, Applications of order preserving operator inequality, Oper. Theory Adv. Appl. 59 (1992), 180-190. MR 94m:47033
  • [6] T. Furuta, Furuta's inequality and its application to the relative operator entropy, J. Operator Theory, 30 (1993), 21-30. MR 95j:47019
  • [7] T. Furuta, Extension of the Furuta inequality and Ando-Hiai log-majorization, Linear Alg. and its Appl., 219 (1995), 139-155. MR 96k:47031
  • [8] T. Furuta, Generalized Aluthge transformation on $p$-hyponormal operators, Proc. Amer. Math. Soc., 124 (1996), 3071-3075. MR 96m:47041
  • [9] E. Heinz, Beiträge zur Störungstheorie der Spektralzerlegung, Math. Ann., 123 (1951), 415-438. MR 13:471f
  • [10] K. Löwner, Über monotone Matrixfunktionen, Math. Z., 38 (1934), 177-216.
  • [11] K. Tanahashi, Best possibility of the Furuta inequality, Proc. Amer. Math. Soc. 124 (1996), 141-146. MR 96d:47025
  • [12] K. Tanahashi, The Furuta inequality with negative power, Proc. Amer. Math. Soc. (to appear). CMP 98:03
  • [13] T. Yoshino, A modified Heinz's inequality, (preprint)

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Additional Information

Kôtarô Tanahashi
Email: tanahasi@tohoku-pharm.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-99-05261-2
Keywords: The L\"owner-Heinz inequality, the Furuta inequality, the grand Furuta inequality
Received by editor(s): September 27, 1997
Received by editor(s) in revised form: March 31, 1998
Published electronically: July 6, 1999
Communicated by: David R. Larson
Article copyright: © Copyright 1999 American Mathematical Society

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