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An Application of Furuta Inequality to Linear Operator Equations

  • Ahn, Eun-Kyung (Department of Mathematics, Kyungpook National University) ;
  • Lim, Yong-Do (Department of Mathematics, Kyungpook National University)
  • Received : 2009.10.09
  • Accepted : 2009.11.28
  • Published : 2009.12.31

Abstract

A class of Hermitian operators B admitting a positive semidefinite solution of the linear operator equation ${\sum}^n_{j=1}A^{n-j}XA^{j-1}=B$ for a fixed positive definite operator A is given via the Furuta inequality.

Keywords

Furuta inequality;L$\"{o}$owner-Heinz inequality;operator equation;Lyapunov equation;positive semidefinite solution

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. https://doi.org/10.1016/j.exmath.2009.02.001
  2. N. Chan and M. Kwong, Hermitian matrix inequalities and a conjecture, Amer. Math. Monthly, 92(1985), 533-541. https://doi.org/10.2307/2323157
  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. https://doi.org/10.1016/j.laa.2009.10.008
  5. M. Kwong, Some results on matrix monotone functions, Linear Alg. and Its Appl., 118(1989), 129-153. https://doi.org/10.1016/0024-3795(89)90577-6