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On a geometric property of positive definite matrices cone. (English) Zbl 1189.15030

Let $A,B\in {ℂ}^{n}$ be positive definite. For $t\in \left[0,1\right]$, the generalized geometric mean $A{#}_{t}B$, of $A$ and $B$ is defined as $A{#}_{t}B={A}^{\frac{1}{2}}{\left({A}^{-\frac{1}{2}}B{A}^{-\frac{1}{2}}\right)}^{t}{A}^{\frac{1}{2}}$. In particular, $A{#}_{\frac{1}{2}}B$ henceforth denoted by $A#B$ is called the geometric mean of $A$ and $B$. Apparently, this notion first appeared in W. Pusz and S. L. Woronowicz [Rep. Math. Phys. 8, 159–170 (1975; Zbl 0327.46032)]. Later, T. Ando [Linear Algebra Appl. 26, 203–241 (1979; Zbl 0495.15018)] developed many of the fundamental properties of the geometric mean in a systematic manner. G. E. Trapp [Linear Multilinear Algebra 16, 113–123 (1984; Zbl 0548.15013)] presented a more recent survey of matrix means. In a recent paper, T. Ando, C.-K. Li and R. Mathias [Linear Algebra Appl. 385, 305–334 (2004; Zbl 1063.47013)] have shown for positive definite matrices $A,B,C,D$ that $A#B=C#D$ implies $\left(A#C\right)#\left(B#D\right)=A#B$.

The authors of the paper under review obtain the following generalization (Theorem 3.1) of the above result: In what follows, $X\le Y$ means that $Y-X$ is positive semi-definite and $0 means that $X$ is positive definite. Suppose that $A,B,C,D$ are positive definite invertible matrices such that $0 for some positive numbers $m$ and $M$. Suppose also that $A{#}_{\alpha }B=C{#}_{\alpha }D=G$ for some $\alpha \in \left(0,1\right)$. Set ${\alpha }_{0}=min\left\{\alpha ,1-\alpha \right\}$ and $h=\frac{M}{m}$. Then for each $\beta \in \left[0,1\right]$, the following two matrix inequalities hold:

${\left\{\frac{{\left({h}^{2-\frac{1}{{\alpha }_{0}}}+1\right)}^{2}}{4{h}^{2-\frac{1}{{\alpha }_{0}}}}\right\}}^{-{\alpha }_{0}}\phantom{\rule{3.33333pt}{0ex}}G\le \left(A{#}_{\beta }C\right){#}_{\alpha }\left(B{#}_{\beta }D\right)\le {\left\{\frac{{\left({h}^{2-\frac{1}{{\alpha }_{0}}}+1\right)}^{2}}{4{h}^{2-\frac{1}{{\alpha }_{0}}}}\right\}}^{-{\alpha }_{0}}G·$

The authors also obtain certain examples and counterexamples related to the main result.

##### MSC:
 15A45 Miscellaneous inequalities involving matrices 15B48 Positive matrices and their generalizations; cones of matrices 26E60 Means