## Osaka Journal of Mathematics

### Midpoints for Thompson's metric on symmetric cones

#### Abstract

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].

#### Article information

Source
Osaka J. Math., Volume 54, Number 1 (2017), 197-208.

Dates
First available in Project Euclid: 3 March 2017

https://projecteuclid.org/euclid.ojm/1488531790

Mathematical Reviews number (MathSciNet)
MR3619754

Zentralblatt MATH identifier
1373.53055

Subjects
Secondary: 15B48: Positive matrices and their generalizations; cones of matrices

#### Citation

Lemmens, Bas; Roelands, Mark. Midpoints for Thompson's metric on symmetric cones. Osaka J. Math. 54 (2017), no. 1, 197--208. https://projecteuclid.org/euclid.ojm/1488531790

#### References

• M. Akian, S. Gaubert, B. Lemmens, and R.D. Nussbaum: Iteration of order preserving subhomogeneous maps on a cone, Math. Proc. Cambridge Philos. Soc. 140 (2006), 157–176.
• C.D. Aliprantis and R. Tourky: Cones and duality, Graduate Studies in Mathematics, 84. American Mathematical Society, Providence, RI, 2007.
• E. Andruchow, G. Corach and D. Stojanoff: Geometrical significance of Löwner-Heinz inequality, Proc. Amer. Math. Soc. 128 (2000), 1031-1037.
• R. Bhatia: Positive definite matrices. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ, 2007.
• G. Corach and A.L. Maestripieri: Differential and metrical structure of positive operators, Positivity 3 (1999), 297–315.
• G. Corach, H. Porta, and L. Recht: Convexity of the geodesic distance on spaces of positive operators, Illinois J. Math. 38 (1994), 87–94.
• J. Faraut and A. Korányi: Analysis on Symmetric Cones, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1994.
• S. Gaubert and Z. Qu: The contraction rate in Thompson's part metric of order-preserving flows on a cone: application to generalized Riccati equations, J. Differential Equations 256 (2014), 2902–2948.
• D.H. Hyers, G. Isac, and T.M. Rassias: Topics in nonlinear analysis & applications. World Scientific Publishing Co., Inc., River Edge, NJ, 1997.
• J. Lawson and Y. Lim: Metric convexity of symmetric cones, Osaka J. Math. 44 (2007), 795–816.
• J. Lawson and Y. Lim: Weighted means and Karcher equations of positive operators, Proc. Natl. Acad. Sci. USA 110 (2013), 5626–5632.
• J. Lawson and Y. Lim: Karcher means and Karcher equations of positive definite operators, Trans. Amer. Math. Soc. Ser. B 1, (2012), 1–22.
• H. Lee and Y. Lim: Carlson's iterative mean algorithm of positive definite matrices, Linear Algebra Appl. 439 (2013), 1183–1201.
• B. Lemmens and R. Nussbaum: Nonlinear Perron-Frobenius theory. Cambridge Tracts in Mathematics 189, Cambridge Univ. Press, Cambridge, 2012.
• B. Lemmens, B. Lins, R. Nussbaum, and M. Wortel: Denjoy-Wolff theorems for Hilbert's and Thompson's metric spaces, J. Anal. Math., to appear.
• B. Lemmens and M. Roelands: Unique geodesics for Thompson's metric, Ann. Inst. Fourier (Grenoble) 65 (2015), 315–348.
• Y. Lim: Finsler metrics on symmetric cones, Math. Ann. 316, (2000), 379–389.
• Y. Lim: Geometry of midpoint sets for Thompson's metric, Linear Algebra Appl., 439 (2013), 211–227.
• Y. Lim and M. Pálfia: The matrix power means and the Karcher mean, J. Funct. Anal. 262, (2012), 1498–1514.
• L. Molnár: Thompson isometries of the space of invertible positive operators, Proc. Amer. Math. Soc. 137 (2009), 3849–3859.
• R.D. Nussbaum: Finsler structures for the part metric and Hilbert's projective metric and applications to ordinary differential equations, Differential Integral Equations 7 (1994), 1649–1707.
• R.D. Nussbaum: Hilbert's projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc. 391,(1988), 1–137.
• M. Pálfia and D.Petz: Weighted multivariable operator means of positive definite operators, Linear Algebra Appl. 463 (2014), 134–153.
• A.C. Thompson: On certain contraction mappings in a partially ordered vector space, Proc. Amer. Math. Soc. 14 (1963), 438–443.