owner-Heinz inequality;operator equation;Lyapunov equation;positive semidefinite solution;"/> owner-Heinz inequality;operator equation;Lyapunov equation;positive semidefinite solution;"/> An Application of Furuta Inequality to Linear Operator Equations | Korea Science
An Application of Furuta Inequality to Linear Operator Equations

• Journal title : Kyungpook mathematical journal
• Volume 49, Issue 4,  2009, pp.743-750
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2009.49.4.743
Title & Authors
An Application of Furuta Inequality to Linear Operator Equations
Ahn, Eun-Kyung; Lim, Yong-Do;

Abstract
A class of Hermitian operators B admitting a positive semidefinite solution of the linear operator equation $\small{{\sum}^n_{j=1}A^{n-j}XA^{j-1}=B}$ for a fixed positive definite operator A is given via the Furuta inequality.
Keywords
Furuta inequality;L$\small{\"{o}}$owner-Heinz inequality;operator equation;Lyapunov equation;positive semidefinite solution;
Language
English
Cited by
References
1.
R. Bhatia and M. Uchiyama, The operator equation ${\sum_{i=0}^{n}A^{n-i}XB^i}$ = Y, Expo. Math., 27(2009), 251-255.

2.
N. Chan and M. Kwong, Hermitian matrix inequalities and a conjecture, Amer. Math. Monthly, 92(1985), 533-541.

3.
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.

4.
T. Furuta, Positive semidefinite solution of the operator equation ${\sum_{j=1}^{n}A^{n-j}XA^{j-1}=B$, Linear Alg. and Its Appl., 432(2010), 949-955.

5.
M. Kwong, Some results on matrix monotone functions, Linear Alg. and Its Appl., 118(1989), 129-153.