Hopf-Rinow theorem in the Sato Grassmanian

ESTEBAN ANDRUCHOW; GABRIEL LAROTONDA

JOURNAL OF FUNCTIONAL ANALYSIS

Academic Press, Elsevier

Lugar: Amsterdam, Holanda; Año: 2008 vol. 255 p. 1692 - 1712

0022-1236

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 { upu*: u in 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)^perp. It is known that the components of Gr_res(p) are differentiable manifolds. Here we give a simple proof of the fact that the identity component 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 theform g(t)=e^{tz}pe^{-tz}, for z a p-codiagonal anti-hermitic element of  B_2(H), have minimal length provided that |z|< pi/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 in 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 the time parameter 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).