A Geometrical Approach to Hilbert-Schmidt Operators
| dc.creator | Larotonda, Gabriel | |
| dc.date | 2008-08-19 | |
| dc.date.accessioned | 2026-07-07T09:57:18Z | |
| dc.date.available | 2026-07-07T09:57:18Z | |
| dc.description | 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 $G_M$, 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 $Π_M:Σ\to M$, a map which is a contraction for the geodesic distance. As a byproduct, we prove the isomorphism $NM\simeqΣ$ (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 $G\simeq M\times \exp(T_1M^{\perp})\times K$. | |
| dc.description | 26 pages, 3 figures | |
| dc.identifier | https://arxiv.org/abs/0808.2524 | |
| dc.identifier | http://arxiv.org/abs/0808.2524 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/167295 | |
| dc.subject | Differential Geometry | |
| dc.subject | Operator Algebras | |
| dc.subject | 22E65 (Primary) 58E50, 53C35, 53C45, 58B20 (Secondary) | |
| dc.title | A Geometrical Approach to Hilbert-Schmidt Operators | |
| dc.type | text |