×

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.

MSC:

47A63 Linear operator inequalities

Citations:

Zbl 0822.15008
PDFBibTeX XMLCite
Full Text: DOI

References:

[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.