Hopf–Rinow theorem in the Sato Grassmannian

Abstract

Let U 2 ( 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 ∈ U 2 ( H ) } , of an infinite projection p in H . This orbit coincides with the connected component of p in the Hilbert–Schmidt restricted Grassmannian Gr res ( 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 Gr res ( p ) are differentiable manifolds. Here we give a simple proof of the fact that Gr res 0 ( p ) is a smooth submanifold of the affine Hilbert space p + B 2 ( H ) , where B 2 ( H ) denotes the space of Hilbert–Schmidt operators of H . Also we show that Gr res 0 ( 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 ) = e t z p e − t z , for z a p-co-diagonal anti-hermitic element of B 2 ( 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 p 1 , p 2 ∈ Gr res 0 ( p ) are joined by a minimal geodesic. If moreover ‖ p 1 − p 2 ‖ < 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 U 2 ( H ) .