Finsler geometry and actions of the p-Schatten unitary groups

Abstract

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 $\mathcal{O}$ be a smooth manifold on which $U_p(H)$ acts transitively and smoothly. Then one can endow $\mathcal{O}$ 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 $$x\in \mathcal{O}andX\in (T\mathcal{O}{)}_x$$ there exists a curve $\delta (t)=e^{tz}\cdot x$ in $\mathcal{O}$, with $z$ a skew-hermitian element in the $p$-Schatten class such that $$\delta (0)=xand\dot{\delta }(0)=X,$$ which remains minimal as long as $t\Vert z{\Vert }_p\le \pi /4$. Moreover, $\delta$ is unique with these properties. We also show that the metric space $(\mathcal{O},d)$ ($d=$ 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 $$\mathcal{O}=\{uA{u}^{\ast }:u\in U_p(H)\}$$ of a self-adjoint operator $A\in B(H)$.

25 pages