The Hadamard product and the Karcher mean of positive invertible operators (Q520743)

Summary: In this paper, we show several operator inequalities involving the Hadamard product and the Karcher mean of \(n (\geq 3)\) positive invertible operators on a separable Hilbert space, which are regarded as an \(n\)-variable operator version of results due to \textit{T. Ando} [Linear Algebra Appl. 26, 203--241 (1979; Zbl 0495.15018); ibid. 223--224, 57--64 (1995; Zbl 0829.15010)] and \textit{J. S. Aujla} and \textit{H. L. Vasudeva} [Math. Japon. 42, No. 2, 265--272 (1995; Zbl 0859.15007)]. As applications, we show estimates from above for an \(n\)-variable version of the Fiedler-type theorem due to \textit{J. I. Fujii} [ibid. 41, No. 3, 531--535 (1995; Zbl 0826.47016)]. Moreover, we show an \(n\)-variable version of the majorization relation due to Ando [loc. cit.] for the Hadamard product via the Karcher mean of \(n (\geq 3)\) positive-definite matrices.