Geometry of $\mathfrak {I}$-Stiefel manifolds

Abstract:

Let $\mathfrak {I}$ be a separable Banach ideal in the space of bounded operators acting in a Hilbert space $\mathcal {H}$ and $\mathcal {U}(\mathcal {H})_{\mathfrak {I}}$ the Banach-Lie group of unitary operators which are perturbations of the identity by elements in $\mathfrak {I}$. In this paper we study the geometry of the unitary orbits \[ \{ U V : U \in \mathcal {U}(\mathcal {H})_{\mathfrak {I}}\}\] and \[ \{ U V W^* : U,W \in \mathcal {U}(\mathcal {H})_{\mathfrak {I}}\},\] where $V$ is a partial isometry. We give a spatial characterization of these orbits. It turns out that both are included in $V + \mathfrak {I}$, and while the first one consists of partial isometries with the same kernel of $V$, the second is given by partial isometries such that their initial projections and $V^*V$ have null index as a pair of projections. We prove that they are smooth submanifolds of the affine Banach space $V + \mathfrak {I}$ and homogeneous reductive spaces of $\mathcal {U}(\mathcal {H})_{\mathfrak {I}}$ and $\mathcal {U}(\mathcal {H})_{\mathfrak {I}} \times \mathcal {U}(\mathcal {H})_{\mathfrak {I}}$ respectively. Then we endow these orbits with two equivalent Finsler metrics, one provided by the ambient norm of the ideal and the other given by the Banach quotient norm of the Lie algebra of $\mathcal {U}(\mathcal {H})_{\mathfrak {I}}$ (or $\mathcal {U}(\mathcal {H})_{\mathfrak {I}} \times \mathcal {U}(\mathcal {H})_{\mathfrak {I}}$) by the Lie algebra of the isotropy group of the natural actions. We show that they are complete metric spaces with the geodesic distance of these metrics.