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On a geometric property of positive definite matrices cone. (English) Zbl 1189.15030
Let $A,B \in \mathbb{C}^n$ be positive definite. For $t \in [0,1]$, the generalized geometric mean $A {\#}_t B$, of $A$ and $B$ is defined as $A {\#}_t B=A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^t A^{\frac{1}{2}}$. 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 {\it W. Pusz} and {\it S. L. Woronowicz} [Rep. Math. Phys. 8, 159--170 (1975; Zbl 0327.46032)]. Later, {\it 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. {\it 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, {\it T. Ando, C.-K. Li} and {\it 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 $(A \# C)\#(B \#D)=A \# B$. The authors of the paper under review obtain the following generalization (Theorem 3.1) of the above result: In what follows, $X \leq Y$ means that $Y-X$ is positive semi-definite and $0<X$ means that $X$ is positive definite. Suppose that $A,B,C,D$ are positive definite invertible matrices such that $0<mI \leq A,B,C,D \leq MI$ for some positive numbers $m$ and $M$. Suppose also that $A {\#}_{\alpha} B=C {\#}_{\alpha} D=G$ for some $\alpha \in (0,1)$. Set ${\alpha}_0= \min\{\alpha, 1-\alpha\}$ and $h=\frac{M}{m}$. Then for each $\beta \in [0,1]$, the following two matrix inequalities hold: $$\left\{ \frac{(h^{2-\frac{1}{{\alpha}_0}} +1)^2}{4h^{2-\frac{1}{{\alpha}_0}}} \right\}^{-{\alpha}_0} ~G \leq (A {\#}_{\beta} C) {\#}_{\alpha}(B {\#}_{\beta} D) \leq \left\{ \frac{(h^{2-\frac{1}{{\alpha}_0}} +1)^2}{4h^{2-\frac{1}{{\alpha}_0}}} \right\}^{-{\alpha}_0}G.$$ The authors also obtain certain examples and counterexamples related to the main result.

15A45Miscellaneous inequalities involving matrices
15B48Positive matrices and their generalizations; cones of matrices
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