Nonpositive curvature: A geometrical approach to Hilbert–Schmidt operators

We give a Riemannian structure to the set Σ of positive invertible unitized Hilbert–Schmidt operators, by means of the trace inner product. This metric makes of Σ a nonpositively curved, simply connected and metrically complete Hilbert manifold. The manifold Σ is a universal model for symmetric spaces of the noncompact type: any such space can be isometrically embedded into Σ. We give an intrinsic algebraic characterization of convex closed submanifolds M  . We study the group of isometries of such submanifolds: we prove that GMGM, the Banach–Lie group generated by M, acts isometrically and transitively on M  . Moreover, GMGM admits a polar decomposition relative to M  , namely GM≃M×KGM≃M×K as Hilbert manifolds (here K   is the isotropy of p=1p=1 for the action Ig:p↦gpg∗), and also GM/K≃MGM/K≃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 ΠM:Σ→M, a map which is a contraction for the geodesic distance. As a byproduct, we prove the isomorphism NM≃ΣNM≃Σ (here NM stands for the normal bundle of a convex closed submanifold M  ). Writing down the factorizations for fixed eaea, we obtain ea=exevexea=exevex with ex∈Mex∈M and v orthogonal to M   at p=1p=1. As a corollary we obtain decompositions for the full group of invertible elements G≃M×exp(T1M⊥)×KG≃M×exp(T1M⊥)×K.