Finsler geometry and actions of the $p$-Schatten unitary groups

Abstract:

Let $p$ be an even positive integer and $U_p(\mathcal {H})$ the Banach-Lie group of unitary operators $u$ which verify that $u-1$ belongs to the $p$-Schatten ideal $\mathcal {B}_p(\mathcal {H})$. Let $\mathcal {O}$ be a smooth manifold on which $U_p(\mathcal {H})$ 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 $U_p(\mathcal {H})$. Our main result establishes that for any pair of given initial conditions \[ x\in \mathcal {O}\hbox { and } X\in (T \mathcal {O})_x \] 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 \[ \delta (0)=x \hbox { and } \dot {\delta }(0)=X, \] 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 $(\mathcal {O},d)$ (where $d$ is the rectifiable distance) is complete. In the process we establish minimality results in the groups $U_p(\mathcal {H})$ and a convexity property for the rectifiable distance. As an example of these spaces, we treat the case of the unitary orbit \[ \mathcal {O}=\{uAu^*: u\in U_p(\mathcal {H})\} \] of a self-adjoint operator $A\in \mathcal {B}(\mathcal {H})$.