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

Let A,B n be positive definite. For t[0,1], the generalized geometric mean A# t B, of A and B is defined as A# t B=A 1 2 (A -1 2 BA -1 2 ) t A 1 2 . In particular, A# 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 (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, XY 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<mIA,B,C,DMI for some positive numbers m and M. Suppose also that A# α B=C# α D=G for some α(0,1). Set α 0 =min{α,1-α} and h=M m. Then for each β[0,1], the following two matrix inequalities hold:

(h 2-1 α 0 +1) 2 4h 2-1 α 0 -α 0 G(A# β C)# α (B# β D)(h 2-1 α 0 +1) 2 4h 2-1 α 0 -α 0 G·

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

MSC:
15A45Miscellaneous inequalities involving matrices
15B48Positive matrices and their generalizations; cones of matrices
26E60Means