Hopf–Rinow theorem in the Sato Grassmannian

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 upu∗: 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 Gr0 res(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 Gr0 res(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) = etzpe−tz, for z a p-co-diagonal anti-hermitic element of B2(H), have minimal length provided that z 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 ∈ Gr0 res(p) are joined by a minimal geodesic. If moreover p1 − p2 < 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).