The Hadamard Product and the Karcher Mean of Positive Invertible Operators

Abstract

In this paper, we show several operator inequalities involving the Hadamard product and the Karcher mean of $n(\ge 3)$ positive invertible operators on a separable Hilbert space, which are regarded as an $n$-variable operator version of results due to Ando and Aujla–Vasudeva. As applications, we show estimates from above for an $n$-variable version of the Fiedler-type theorem due to Fujii. Moreover, we show an $n$-variable version of the majorization relation due to Ando for the Hadamard product via the Karcher mean of $n(\ge 3)$ positive-definite matrices.