Abstract. Let p be an even positive integer and U p (H) the Banach-Lie group of unitary operators u which verify that u − 1 belongs to the p-Schatten ideal B p (H). Let O be a smooth manifold on which U p (H) acts transitively and smoothly. Then one can endow O with a natural Finsler metric in terms of the p-Schatten norm and the action of U p (H). Our main result establishes that for any pair of given initial conditionsthere exists a curve δ(t) = e tz · x in O, with z a skew-hermitian element in the p-Schatten class, such thatwhich remains minimal as long as t z p ≤ π/4. Moreover, δ is unique with these properties. We also show that the metric space (O, d) (where d is the rectifiable distance) is complete. In the process we establish minimality results in the groups U p (H) and a convexity property for the rectifiable distance. As an example of these spaces, we treat the case of the unitary orbit O = {uAu
Let U 2 (H) be the Banach-Lie group of unitary operators in the Hilbert space H which are HilbertSchmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit upu * : u ∈ U 2 (H) , of an infinite projection p in H. This orbit coincides with the connected component of p in the HilbertSchmidt 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 0 res (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 0 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) = e tz pe −tz , 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 0 res (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).
Let U c (H) = {u : u unitary and u − 1 compact} stand for the unitary Fredholm group. We prove the following convexity result. Denote by d ∞ the rectifiable distance induced by the Finsler metric given by the operator norm in U c (H). If u 0 , u 1 , u ∈ U c (H) and the geodesic β joining u 0 and u 1 inIn particular the convexity radius of the geodesic balls in U c (H) is π/4. The same convexity property holds in the p-Schatten unitary groups U p (H) = {u : u unitary and u − 1 in the p-Schatten class}, for p an even integer, p ≥ 4 (in this case, the distance is strictly convex). The same results hold in the unitary group of a C * -algebra with a faithful finite trace. We apply this convexity result to establish the existence of curves of minimal length with given initial conditions, in the unitary orbit of an operator, under the action of the Fredholm group. We characterize self-adjoint operators A such that this orbit is a submanifold (of the affine space A + K(H), where K(H)=compact operators).
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