The inverse problem for generalized contraharmonic means

T. H. Dinh ; C. T. Le ; B. K. Vo

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2022), pp. 3-9

Résumé

In this paper we introduce the generalized contraharmanic mean associated to a Kubo-Ando mean $\sigma$ as $$ C_\sigma(X, Y) = X\sigma Y - X\sigma^\perp Y, $$ where $\sigma^\perp$ is the dual mean of $\sigma$ and $X, Y$ are positive definite matrices. We show that for a symmetric Kubo-Ando mean $\sigma$ such as $\sigma \ge \sharp$ and for any positive definite matrices $A \ge B$ the inverse problem \begin{equation*} A=C_\sigma(X, Y), \ \ B=X^{1/2}(X^{-1/2}YX^{-1/2})^{1/2}X^{1/2} \end{equation*} has a positive solution $(X, Y)$.

Détail

Keywords: Kubo-Ando means, geometric mean, generalized contraharmonic mean, inverse problem, Brouwer's fixed point theorem, non-linear matrix equations.