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 ${\cal O}$ 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 $\delta(t)=e^{tz}\cdot x$ in ${\cal O}$, with $z$ a skew-hermitian element in the $p$-Schatten class such that $$ \delta(0)=x \hbox{and} \dot{\delta}(0)=X, $$ which remains minimal as long as $t\|z\|_p\le \pi/4$. Moreover, $\delta$ 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 $A\in B(H)$.