Nonpositive Curvature: A Geometric Approach to Hilbert-Schmidt Operators

GABRIEL LAROTONDA

DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

Elsevier

Año: 2007 vol. 25 p. 679 - 700

0926-2245

We give a Riemannian structure to the set $Sigma$ of positive invertible unitized Hilbert–Schmidt operators, by means of the trace inner product. This metric makes of $Sigma$ a nonpositively curved, simply connected and metrically complete Hilbert manifold. This manifold is a universal model for symmetric spaces of the noncompact type: any such space can be isometrically embedded into $Sigma$. We give an intrinsic algebraic characterization of convex closed submanifolds M. We study the group of isometries of such submanifolds: we prove that $G_M$ , the Banach–Lie group generated by M, acts isometrically and transitively on M. Moreover, $G_M$ admits a polar decomposition relative to M, namely $G_M=G x K$ as Hilbert manifolds (here K is the isotropy of p=1 for the action $I_g:pmapsto gpg^*$), and also $G_M/Kcap G_M=M$ so M is an homogeneous space. We obtain several decomposition theorems by means of geodesically convex submanifolds M. These decompositions are obtained via a nonlinear but analytic orthogonal  projection $Pi_M: Sigma o M$, a map which is a contraction for the geodesic distance. As a byproduct, we prove the isomorphism $NM=Sigma$ (here NM stands for the normal bundle of a convex closed submanifold M). Writing down the factorizations for fixed $e^a$ , we obtain $e^a = e^x e^v e^x$ with $e^x in M$ and v orthogonal to M at p = 1. As a corollary we obtain decompositions for the full group of invertible elements $G=G_M x exp(T_1M^{perp}) x K$.