Finsler geometry and actions of the p-Schatten unitary groups
| dc.creator | Andruchow, Esteban | |
| dc.creator | Larotonda, Gabriel | |
| dc.creator | Recht, Lazaro | |
| dc.date | 2008-08-16 | |
| dc.date.accessioned | 2026-07-07T09:57:04Z | |
| dc.date.available | 2026-07-07T09:57:04Z | |
| dc.description | Let $p$ be an even positive integer and $U_p(H)$ be the Banach-Lie group of unitary operators $u$ which verify that $u-1$ belongs to the $p$-Schatten ideal $B_p(H)$. Let ${\cal O}$ be a smooth manifold on which $U_p(H)$ acts transitively and smoothly. Then one can endow ${\cal O}$ with a natural Finsler metric in terms of the $p$-Schatten norm and the action of $U_p(H)$. Our main result establishes that for any pair of given initial conditions $$ x\in {\cal O}\hbox{and} X\in (T{\cal O})_x $$ there exists a curve $δ(t)=e^{tz}\cdot x$ in ${\cal O}$, with $z$ a skew-hermitian element in the $p$-Schatten class such that $$ δ(0)=x \hbox{and} \dotδ(0)=X, $$ which remains minimal as long as $t\|z\|_p\le π/4$. Moreover, $δ$ is unique with these properties. We also show that the metric space $({\cal O},d)$ ($d=$ rectifiable distance) is complete. In the process we establish minimality results in the groups $U_p(H)$, and a convexity property for the rectifiable distance. As an example of these spaces, we treat the case of the unitary orbit $$ {\cal O}=\{uAu^*: u\in U_p(H)\} $$ of a self-adjoint operator $A\in B(H)$. | |
| dc.description | 25 pages | |
| dc.identifier | https://arxiv.org/abs/0808.2274 | |
| dc.identifier | http://arxiv.org/abs/0808.2274 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/167206 | |
| dc.subject | Differential Geometry | |
| dc.subject | Operator Algebras | |
| dc.subject | 22E65 (Primary) 58E50, 58B20 (Secondary) | |
| dc.title | Finsler geometry and actions of the p-Schatten unitary groups | |
| dc.type | text |