Lagrangian Grassmannian in infinite dimension

JOURNAL OF GEOMETRY AND PHYSICS

卷:59

Article

Andruchow, Esteban¹  Larotonda, Gabriel¹ 

[1] Univ Nacl Gen Sarmiento JM Gutierrez, Inst Ciencias, RA-1150 Los Polvorines, Argentina

关键词: Complex structure;    Lagrangian subspace;    Short geodesic;   

DOI  :  10.1016/j.geomphys.2008.11.004

来源: Elsevier

【 摘 要 】

Given a complex structure J on a real (finite or infinite dimensional) Hilbert space H, we study the geometry of the Lagrangian Grassmannian Lambda(H) of it. i.e. the set of closed linear subspaces L subset of H such that J(L) = L-perpendicular to. The complex unitary group U(H-J), consisting of the elements of the orthogonal group of H which are complex linear for the given complex structure, acts transitively on Lambda(H) and induces a natural linear connection in Lambda(H). It is shown that any pair of Lagrangian subspaces can be joined by a geodesic of this connection. A Finsler metric can also be introduced. if one regards subspaces L as projections p(L) (=the orthogonal projection onto L) or symmetries epsilon(L) = 2p(L) - 1, namely measuring tangent vectors with the operator norm. We show that for this metric the Hopf-Rinow theorem is valid in Lambda(M): a geodesic joining a pair of Lagrangian subspaces can be chosen to be of minimal length. A similar result holds for the unitary orbit of a Lagrangian subspace under the action of the k-Schatten unitary group (2 <= k <= infinity), with the Finsler metric given by the k-norm. (C) 2008 Elsevier B.V. All rights reserved.

【 授权许可】

Free