Symmetric spaces with convex metrics

DOI10.1515/FORUM.2007.023zbMath1169.53333OpenAlexW2003737008MaRDI QIDQ5421640

Yongdo Lim, Jimmie D. Lawson

Publication date: 24 October 2007

Published in: Forum Mathematicum (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1515/forum.2007.023

zbMATH Keywords

\(C^*\)-algebralineated symmetric spacespositive Hermitian elements

Mathematics Subject Classification ID

Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds (58B20) Differential geometry of symmetric spaces (53C35) Global differential geometry of Finsler spaces and generalizations (areal metrics) (53C60) Pseudogroups and differentiable groupoids (58H05)

Related Items (28)

Weighted geometric means ⋮ Solving symmetric matrix word equations via symmetric space machinery ⋮ Weighted geometric mean of \(n\)-operators with \(n\)-parameters ⋮ On the nonlinear matrix equation X^p = A#_t(M^T (X^–1 + B)^–1 M) ⋮ Barycentric maps for compactly supported measures ⋮ Homogeneous Sylvester equations of lower triangular matrices ⋮ Mostow's decomposition theorem for \(L^\ast\)-groups and applications to affine coadjoint orbits and stable manifolds ⋮ Stopping criteria for the Ando-Li-Mathias and Bini-Meini-Poloni geometric means ⋮ A binomial expansion formula for weighted geometric means of unipotent matrices ⋮ Multi-variable weighted geometric means of positive definite matrices ⋮ Existence, uniqueness and an ODE approach to the \(L^1\)-Karcher mean ⋮ Unnamed Item ⋮ Matrix power means and the Karcher mean ⋮ On the nonlinear matrix equation \(X^p = A + M^T (X \# B) M\) ⋮ Metric convexity of symmetric cones ⋮ On Ando-Li-Mathias geometric mean equations ⋮ \(\varGamma \)-commuting triples of positive definite matrices and midpoint equations on unitary matrices ⋮ A converse inequality of higher order weighted arithmetic and geometric means of positive definite operators ⋮ Existence and uniqueness of the Karcher mean on unital \(C^\ast\)-algebras ⋮ Convergence theorems for barycentric maps ⋮ Moving averages on convex metric spaces ⋮ Hilbert and Thompson isometries on cones in JB-algebras ⋮ Hilbert isometries and maximal deviation preserving maps on JB-algebras ⋮ Bushell's equations and polar decompositions ⋮ A geometric path from zero Lyapunov exponents to rotation cocycles ⋮ Loos symmetric cones ⋮ Smooth Bruck loops, symmetric spaces, and nonassociative vector spaces ⋮ Strong law of large numbers for the \(L^1\)-Karcher mean

Cites Work

• On the positive definite solutions of the matrix equations \(X^{s}\pm A^{\text T} X^{-t} A=I_{n}\)
• Symmetric incidence groupoids
• Means on dyadic symmetric sets and polar decompositions
• Recent developments on absolute geometries and algebraization by K-loops
• A Cartan-Hadamard theorem for Banach-Finsler manifolds
• Finsler metrics on symmetric cones
• The geometry of the space of selfadjoint invertible elements in a \(C^*\)-algebra
• Spaces with non-positive curvature
• Eigenvalues of Words in Two Positive Definite Letters
• Iterative solution of two matrix equations
• The Geometric Mean, Matrices, Metrics, and More
• Symmetric word equations in two positive definite letters
• Efficient computation of the extreme solutions of $X+A^*X^{-1}A=Q$ and $X-A^*X^{-1}A=Q$
• Geometrical significance of the Löwner-Heinz inequality

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