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January 2017 Midpoints for Thompson's metric on symmetric cones
Bas Lemmens, Mark Roelands
Osaka J. Math. 54(1): 197-208 (January 2017).


We characterise the affine span of the midpoints sets, $\mathcal{M}(x,y)$, for Thompson's metric on symmetric cones in terms of a translation of the zero-component of the Peirce decomposition of an idempotent. As a consequence we derive an explicit formula for the dimension of the affine span of $\mathcal{M}(x,y)$ in case the associated Euclidean Jordan algebra is simple. In particular, we find for $A$ and $B$ in the cone positive definite Hermitian matrices that \[ \dim({\rm aff}\, \mathcal{M}(A,B))=q^2, \] where $q$ is the number of eigenvalues $\mu$ of $A^{-1}B$, counting multiplicities, such that \[ \mu\neq \max\{\lambda_+(A^{-1}B),\lambda_-(A^{-1}B)^{-1}\}, \] where $\lambda_+(A^{-1}B):=\max \{\lambda\colon \lambda\in\sigma(A^{-1}B)\}$ and $\lambda_-(A^{-1}B):=\min\{\lambda\colon \lambda\in\sigma(A^{-1}B)\}$. These results extend work by Y. Lim [18].


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Bas Lemmens. Mark Roelands. "Midpoints for Thompson's metric on symmetric cones." Osaka J. Math. 54 (1) 197 - 208, January 2017.


Published: January 2017
First available in Project Euclid: 3 March 2017

zbMATH: 1373.53055
MathSciNet: MR3619754

Primary: 53C22
Secondary: 15B48

Rights: Copyright © 2017 Osaka University and Osaka City University, Departments of Mathematics


Vol.54 • No. 1 • January 2017
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