A proof of an order preserving inequality. (English) Zbl 1010.47014

From the introduction: We obtained the following result for bounded linear operators on a Hilbert space in [T. Furuta, Linear Algebra Appl. 219, 139-155 (1995; Zbl 0822.15008)].
Theorem A. If \(A\geq B>0\), then for each \(t\in [0,1]\) and \(p\geq 1\) \[ A^{1+r-t}\geq \{A^{\frac r2}(A^{\frac{-t}{2}}B^pA^{\frac{-t}{2}})^sA^{\frac r2}\}^{\frac{1+r-t}{(p-t)s+r}}\tag{1} \] holds for \(r\geq t\) and \(s\geq 1\).
M. Uchiyama [J. Math. Soc. Japan 55, 197-207 (2003)] showed the following interesting extension of Theorem A.
Theorem B. If \(A\geq B\geq C>0\), then for each \(t\in [0,1]\) and \(p\geq 1\) \[ A^{1+r-t}\geq \{A^{\frac r2}(B^{\frac{-t}{2}}C^p B^{\frac{-t}2})^s A^{\frac r2}\}^{\frac{1+r-t}{(p-t)s+r}}\tag{2} \] holds for \(r\geq t\) and \(s\geq 1\).
Here we show a simplified proof of Theorem B by using Theorem A itself.


47A63 Linear operator inequalities


Zbl 0822.15008
Full Text: DOI


[1] Furuta, T.: Extension of the Furuta inequality and Ando-Hiai log majorization. Linear Algebra and Its Appl., 219 , 139-155 (1995). · Zbl 0822.15008 · doi:10.1016/0024-3795(93)00203-C
[2] Furuta, T.: Invitation to Linear Operators. Taylor & Francis, London (2001). · Zbl 1029.47001
[3] Uchiyama, M.: Criteria for monotonicity of operator means. J. Math. Soc. Japan. (To appear). · Zbl 1036.47008 · doi:10.2969/jmsj/1196890849
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