Proceedings of the American Mathematical Society

Author(s): Takayuki Furuta; Derming Wang
Journal: Proc. Amer. Math. Soc. 126 (1998), 2427-2432.
MSC (1991): Primary 47A63
Retrieve article in: PDF
This article is available free of charge

Abstract: Let $A\ge B\ge 0$ with $A>0$

and let $t\in[0,1]$ and $q\ge 0$ . As a generalization of a result due to Furuta, it is shown that the operator function

$\begin{displaymath}G_{p,q,t}(A,B,r,s)=A^{-r/2}\{A^{r/2} (A^{-t/2} B^pA^{-t/2})^s A^{r/2}\}^{(q-t+r)/[(p-t)s+r]}A^{-r/2} \end{displaymath}$

is decreasing for $r\ge t$ and $s\ge 1$ if $p\ge\max\{q,t\}$ . Moreover, if $1\ge p>t$ and $q\ge t$ , then $G_{p,q,t}(A,B,r,s)$ is decreasing for $r\ge 0$ and $s\ge \frac{q-t}{p-t}$ . The latter result is an extension of an earlier result of Furuta.

1.: T. Ando and F. Hiai, Log-majorization and complementary Golden-Thompson type inequalities, Linear Alg. and Its Appl. 197, 198 (1994), 113-131. MR 95d:15006
2.: M. Fujii, Furuta's inequality and its mean theoretic approach, J. Operator Theory 23 (1990), 67-72. MR 91g:47012
3.: M. Fujii,T. Furuta and E. Kamei, Furuta's inequality and its application to Ando's theorem, Linear Alg. and Its Appl. 179 (1993), 161-169. MR 93j:47026
4.: M. Fujii and E. Kamei, Mean theoretic approach to the grand Furuta inequality, Proc. Amer. Math. Soc. 124 (1996), 2751-2756. MR 96k:47032
5.: T. Furuta, $A\ge B\ge 0$ assures $(B^rA^p B^r)^{1/q}\ge B^{(p+2r)/q}$ for $r\ge 0$ , $p\ge 0$ , $q\ge 1$ with $(1+2r)q\ge p+2r$ , Proc. Amer. Math. Soc. 101 (1987), 85-88. MR 89b:47028
6.: T. Furuta, A proof via operator means of an order preserving inequality, Linear Alg. and Its Appl. 113(1989), 129-130. MR 89k:47023
7.: T. Furuta, Elementary proof of an order preserving inequality, Proc. Japan Acad. 65 (1989), 126. MR 90g:47029
8.: T. Furuta, Two operator functions with monotone property, Proc. Amer. Math. Soc. 111 (1991), 511-516. MR 91f:47023
9.: T. Furuta, Applications of order preserving operator inequalities, Operator Theory: Advances and Applications 59 (1992), 180-190. MR 94m:47033
10.: T. Furuta, Extension of the Furuta inequality and Ando-Hiai log-majorization, Linear Alg. and Its Appl. 219 (1995), 139-155. MR 96k:47031
11.: T. Furuta, Parallelism related to the inequality `` $A \ge B\ge 0$ ensures $(A^{r/2}A^p A^{r/2})^{(1+r)/(p+r)}\ge (A^{r/2}B^p A^{r/2})^{(1+r)/(p+r)}$ for $p\ge 1$ and $r\ge 0$ '', Math. Japon. 45 (1997), 203-209. MR 98b:47024
12.: F. Hansen, An operator inequality, Math. Ann. 246 (1980), 249-250. MR 82a:46065
13.: E. Heinz, Beiträge zur Störungstheorie der Spektralzerlegung, Math. Ann. 123 (1951), 415-438. MR 13:471f
14.: E. Kamei, A satellite to Furuta's inequality, Math. Japon 33 (1988), 883-886. MR 89m:47011
15.: K. Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), 177-216.
16.: K. Tanahashi, Best possibility of the Furuta inequality, Proc. Amer. Math. Soc. 124 (1996), 141-146. MR 96d:47025

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47A63

Takayuki Furuta
Affiliation: Department of Applied Mathematics, Faculty of Science, Science University of Tokyo, Kagurazaka, Shinjuku 162-8601, Tokyo, Japan
Email: furuta@rs.kagu.sut.ac.jp

Derming Wang
Affiliation: Department of Mathematics, California State University, Long Beach, Long Beach, California 90840-1001

DOI: 10.1090/S0002-9939-98-04632-2
PII: S 0002-9939(98)04632-2
Keywords: L\"owner-Heinz inequality, Furuta inequality
Received by editor(s): January 23, 1997
Communicated by: Palle E. T. Jorgensen
Copyright of article: Copyright 1998, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS