The inverse problem for generalized contraharmonic means
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2022), pp. 3-9
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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 $(X, Y)$.
Keywords:
Kubo-Ando means, geometric mean, generalized contraharmonic mean, inverse problem, Brouwer's fixed point theorem, non-linear matrix equations.
@article{IVM_2022_7_a0,
author = {T. H. Dinh and C. T. Le and B. K. Vo},
title = {The inverse problem for generalized contraharmonic means},
journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
pages = {3--9},
publisher = {mathdoc},
number = {7},
year = {2022},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/IVM_2022_7_a0/}
}
T. H. Dinh; C. T. Le; B. K. Vo. The inverse problem for generalized contraharmonic means. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2022), pp. 3-9. http://geodesic.mathdoc.fr/item/IVM_2022_7_a0/