Geometry of the left action of the p-Schatten groups

Abstract

Let $\mathcal{H}$ be an infinite dimensional Hilbert space, $\mathcal{B}_{p}\left(\mathcal{H}\right)$ the $p$-Schatten class of $\mathcal{H}$ and $U_{p}\left(\mathcal{H}\right)$ be the Banach-Lie group of unitary operators which are $p$-Schatten perturbations of the identity. Let $A$ be a bounded selfadjoint operator in $\mathcal{H}$. We show that $$\mathcal{O}_A:=\left\{UA : U \in U_{p}\left(\mathcal{H}\right) \right\}$$ is a smooth submanifold of the affine space $A + \mathcal{B}_{p}\left(\mathcal{H}\right)$ if only if $A$ has closed range. Furthermore, it is a homogeneous reductive space of $U_{p}\left(\mathcal{H}\right)$. We introduce two metrics: one via the ambient Finsler metric induced as a submanifold of $A + \mathcal{B}_{p}\left(\mathcal{H}\right)$, the other, by means of the quotient Finsler metric provided by the homogeneous space structure. We show that $\mathcal{O}_A$ is a complete metric space with the rectifiable distance of these metrics.