Abstract
This note aims to generalize the reverse weighted arithmetic–geometric mean inequality of n positive invertible operators due to Lawson and Lim. In addition, we make comparisons between the weighted Karcher mean and Lawson–Lim geometric mean for higher powers.
Highlights
Let B(H) be the C∗-algebra of all bounded linear operators on a complex separable Hilbert space H
P stands for the convex cone of positive invertible operators. n denotes the simplex of positive probability vectors in Rn convexly spanned by the unit coordinate vectors. · and | · | denote the operator norm and the unitarily invariant norm, respectively
Lawson and Lim [12] established a definition of the weighted version of the Ando–Li–Mathias geometric mean for n positive operators, we call it Lawson–Lim geometric mean G[n, t](A1, A2, . . . , An); see [12] for more details
Summary
Let B(H) be the C∗-algebra of all bounded linear operators on a complex separable Hilbert space H. Definition 1.1 (Ando–Li–Mathias geometric mean [2]) Let Ai (ii) Assume that the geometric mean of any n – 1-tuple of operators is defined.
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