On a geometric property of positive definite matrices cone
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 \textit{W. Pusz} and \textit{S. L. Woronowicz} [Rep. Math. Phys. 8, 159--170 (1975; Zbl 0327.46032)]. Later, \textit{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. \textit{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, \textit{T. Ando, C.-K. Li} and \textit{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.
scientific article; zbMATH DE number 5702387