Hopf-Rinow theorem in the Sato Grassmannian

Abstract

Let U2 (H) be the Banach-Lie group of unitary operators in the Hilbert space H which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit{u p u* : u ∈ U2 (H)}, of an infinite projection p in H. This orbit coincides with the connected component of p in the Hilbert-Schmidt restricted Grassmannian Grres (p) (also known in the literature as the Sato Grassmannian) corresponding to the polarization H = p (H) ⊕ p (H)⊥. It is known that the components of Grres (p) are differentiable manifolds. Here we give a simple proof of the fact that Grres0 (p) is a smooth submanifold of the affine Hilbert space p + B2 (H), where B2 (H) denotes the space of Hilbert-Schmidt operators of H. Also we show that Grres0 (p) is a homogeneous reductive space. We introduce a natural metric, which consists in endowing every tangent space with the trace inner product, and consider its Levi-Civita connection. This connection has been considered before, for instance its sectional curvature has been computed. We show that the Levi-Civita connection coincides with a linear connection induced by the reductive structure, a fact which allows for the easy computation of the geodesic curves. We prove that the geodesics of the connection, which are of the form γ (t) = et z p e- t z, for z a p-co-diagonal anti-hermitic element of B2 (H), have minimal length provided that {norm of matrix} z {norm of matrix} ≤ π / 2. Note that the condition is given in terms of the usual operator norm, a fact which implies that there exist minimal geodesics of arbitrary length. Also we show that any two points p1, p2 ∈ Grres0 (p) are joined by a minimal geodesic. If moreover {norm of matrix} p1 - p2 {norm of matrix} < 1, the minimal geodesic is unique. Finally, we replace the 2-norm by the k-Schatten norm (k > 2), and prove that the geodesics are also minimal for these norms, up to a critical value of t, which is estimated also in terms of the usual operator norm. In the process, minimality results in the k-norms are also obtained for the group U2 (H).