2008/08/19 by Esteban Andruchow, Andruchow, Esteban, Gabriel Larotonda +1
Mathematics · Physics and Astronomy · #22E65 (Primary) 58E50 #58B20 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Nonlinear Waves and Solitons #Operator Algebras (math.OA) #Quantum Mechanics and Non-Hermitian Physics #math.DG #math.OA #msc:22E65 #msc:58B20 #msc:58E50
paper · pdf · doi:10.48550/arxiv.0808.2525
20 pages
arxiv created 2008/08/19 · openalex publication_date 2008/08/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let U2(\cal H) be the Banach-Lie group of unitary operators in the Hilbert space \cal H which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit \upu^*: u∈ U2(\cal H)\, of an infinite projection p in \cal 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 \cal H=p(\cal H)⊕ p(\cal 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+\cal B2(\cal H), where \cal B2(\cal H) denotes the space of Hilbert-Schmidt operators of \cal H. We prove that the geodesics of the natural connection, which are of the form γ(t)=etzpe-tz, for z a p-codiagonal anti-hermitic element of \cal B2(\cal 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∈ Grres0(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(\cal H).