Non Positively Curved Metric in the Space of Positive Definite Infinite Matrices

we introduce a riemannian metric with non positive curvature in the (infinite dimensional) manifold of positive invertible operators of a hilbert space , which are scalar perturbations of hilbert-schmidt operators. the (minimal) geodesics and the geodesic distance are computed. it is shown that this metric, which is complete, generalizes the well known non positive metric for positive definite complex matrices. moreover, these spaces of finite matrices are naturally imbedded in .