MIDPOINTS FOR THOMPSON'S METRIC ON SYMMETRIC CONES

We characterise the affine span of the midpoints sets, 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 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(aff M(A, B)) = q(2), where q is the number of eigenvalues mu of A(-1)B, counting multiplicities, such that mu not equal max{lambda(+)(A(-1)B), lambda(-)(A(-1)B)(-1)}, where lambda(+)(A(-1)B) := max{lambda: lambda is an element of sigma(A(-1)B)} and lambda(-)(A(-1)B) := min{lambda: lambda is an element of sigma(A(-1)B)}. These results extend work by Y. Lim [18].