A Geometrical Approach to Hilbert-Schmidt Operators

TL;DR

This paper develops a Riemannian geometric framework for positive invertible Hilbert-Schmidt operators, revealing their structure as a universal model for symmetric spaces and establishing various decompositions and isometries.

Contribution

It introduces a novel Riemannian structure on the space of positive invertible Hilbert-Schmidt operators, characterizing its geometric and algebraic properties and deriving new decomposition theorems.

Findings

The space forms a nonpositively curved, complete Hilbert manifold.

Any symmetric space of noncompact type embeds isometrically into this manifold.

Derived several decomposition theorems using geodesically convex submanifolds.

Abstract

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. The manifold $\Sigma$ 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 $G_M\simeq M\times K$ as Hilbert manifolds (here $K$ is the isotropy of $p=1$ for the action $I_g: p\mapsto gpg^*$), and also $G_M/K\simeq M$ so $M$ is an homogeneous space. We obtain several decomposition theorems by means of geodesically convex submanifolds $M$. These decompositions are obtained \textit{via} a nonlinear but analytic orthogonal projection $\Pi_M:\Sigma\to M$, a map which is a contraction for the geodesic distance. As a byproduct, we prove the isomorphism $NM\simeq\Sigma$ (here $NM$ stands for the normal bundle of a convex closed submanifold $M$). Writing down the factorizations for fixed ${\rm e}^a$, we obtain ${\rm e}^a={\rm e}^x{\rm e}^v{\rm e}^x$ with ${\rm e}^x\in M$ and $v$ orthogonal to $M$ at $p=1$. As a corollary we obtain decompositions for the full group of invertible elements $G\simeq M\times \exp(T_1M^{\perp})\times K$.