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The Furuta inequality with negative powers

Author(s): Kôtarô Tanahashi
Journal: Proc. Amer. Math. Soc. 127 (1999), 1683-1692.
MSC (1991): Primary 47B15
Posted: February 11, 1999
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Abstract: Let $ A, B \in B(H)$ be bounded linear operators on a Hilbert space $H$ satisfying $O\leq B\leq A$. Furuta showed the operator inequality $ (A^{r}B^{p}A^{r})^{\frac{1}{q}}\leq $
$ A^{\frac{p+2r}{q}} $ as long as positive real numbers $p , q, r $ satisfy $ p+2r\leq (1+2r)q$ and $ 1\leq q$. In this paper, we show this inequality is valid if negative real numbers $ p, q, r $ satisfy a certain condition. Also, we investigate the optimality of that condition.

References:

[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, Two operator functions with monotone property, Proc. Amer. Math. Soc. 111 (1991), 511-516. MR 91f:47023

[6]
T. Furuta, Applications of order preserving operator inequality, Oper. Theory Adv. Appl. 59 (1992), 180-190. MR 94m:47033

[7]
T. Furuta, Furuta's inequality and its application to the relative operator entropy, J. Operator Theory 30 (1993), 21-30. MR 95j:47019

[8]
T. Furuta, Extension of the Furuta inequality and Ando-Hiai log-majorization, Linear Alg. and its Appl. 219 (1995), 139-155. MR 96k:47031

[9]
T. Furuta, Generalized Aluthge transformation on $p$-hyponormal operators, Proc. Amer. Math. Soc. 124 (1996), 3071-3075. MR 96m:47041

[10]
E. Heinz, Beiträge zur Störungstheorie der Spektralzerlegung, Math. Ann. 123 (1951), 415-438. MR 13:471f

[11]
K. Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), 177-216.

[12]
A. W. Marshall and I. Olkin, Inequalities, Theory of Majorization and Its Applications, Academic Press (1979). MR 81b:00002

[13]
K. Tanahashi, Best possibility of the Furuta inequality, Proc. Amer. Math. Soc. 124 (1996), 141-146. MR 96d:47025

[14]
T. Yoshino, A modified Heinz's inequality, (preprint).

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Additional Information:

Kôtarô Tanahashi
Affiliation: Department of Mathematics, Tohoku College of Pharmacy, Komatsushima, Aoba-ku, Sendai 981, Japan
Email: tanahasi@tohoku-pharm.ac.jp

DOI: 10.1090/S0002-9939-99-04705-X
PII: S 0002-9939(99)04705-X
Keywords: L\"{o}wner-Heinz inequality, the Furuta inequality, positive operator
Received by editor(s): September 29, 1995
Received by editor(s) in revised form: June 12, 1996, August 6, 1996, October 23, 1996, April 3, 1997, and September 4, 1997
Posted: February 11, 1999
Communicated by: Palle E. T. Jorgensen
Copyright of article: Copyright 1999, American Mathematical Society

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