Hopf-Rinow Theorem in the Sato Grassmannian
| dc.creator | Andruchow, Esteban | |
| dc.creator | Larotonda, Gabriel | |
| dc.date | 2008-08-19 | |
| dc.date.accessioned | 2026-07-07T09:57:19Z | |
| dc.date.available | 2026-07-07T09:57:19Z | |
| dc.description | Let $U_2({\cal H})$ be the Banach-Lie group of unitary operators in the Hilbert space ${\cal H}$ which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit $$\{upu^*: u\in U_2({\cal H})\},$$ of an infinite projection $p$ in ${\cal H}$. This orbit coincides with the connected component of $p$ in the Hilbert-Schmidt restricted Grassmannian $Gr_{res}(p)$ (also known in the literature as the Sato Grassmannian) corresponding to the polarization ${\cal H}=p({\cal H})\oplus p({\cal H})^\perp$. It is known that the components of $Gr_{res}(p)$ are differentiable manifolds. Here we give a simple proof of the fact that $Gr_{res}^0(p)$ is a smooth submanifold of the affine Hilbert space $p+{\cal B}_2({\cal H})$, where ${\cal B}_2({\cal H})$ denotes the space of Hilbert-Schmidt operators of ${\cal H}$. We prove that the geodesics of the natural connection, which are of the form $γ(t)=e^{tz}pe^{-tz}$, for $z$ a $p$-codiagonal anti-hermitic element of ${\cal B}_2({\cal H})$, have minimal length provided that $\|z\|\le π/2$. Note that the condition is given in terms of the usual operator norm, a fact which implies that there exist minimal geodesics of arbitrary length. Also we show that any two points $p_1,p_2\in Gr_{res}^0(p)$ are joined by a minimal geodesic. If moreover $\|p_1-p_2\|<1$, the minimal geodesic is unique. Finally, we replace the 2-norm by the $k$-Schatten norm ($k>2$), and prove that the geodesics are also minimal for these norms, up to a critical value of $t$, which is estimated also in terms of the usual operator norm. In the process, minimality results in the $k$-norms are also obtained for the group $U_2({\cal H})$. | |
| dc.description | 20 pages | |
| dc.identifier | https://arxiv.org/abs/0808.2525 | |
| dc.identifier | http://arxiv.org/abs/0808.2525 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/167296 | |
| dc.subject | Differential Geometry | |
| dc.subject | Operator Algebras | |
| dc.subject | 22E65 (Primary) 58E50, 58B20 (Secondary) | |
| dc.title | Hopf-Rinow Theorem in the Sato Grassmannian | |
| dc.type | text |