Non positive metric in the space of positive definite infinite matrices

E. ANDRUCHOW, A. VARELA

REVISTA DE LA UNIóN MATEMáTICA ARGENTINA

Unión Matemática Argentina

Lugar: Bahía Blanca; Año: 2007 vol. 48 p. 7 - 15

0041-6932

We introduce a Riemannian metric with non positive curvature in the (in nite dimensional) manifold 1 of positive invertible operators of a Hilbert space H, 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 de nite complex matrices. Moreover, these spaces of nite matrices are naturally imbedded in 1.1 of positive invertible operators of a Hilbert space H, 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 de nite complex matrices. Moreover, these spaces of nite matrices are naturally imbedded in 1.H, 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 de nite complex matrices. Moreover, these spaces of nite matrices are naturally imbedded in 1.1.