Lagrangian Grassmannian in infinite dimension

Abstract

Given a complex structure J on a real (finite or infinite dimensional) Hilbert space H, we study the geometry of the Lagrangian Grassmannian Λ (H) of H, i.e. the set of closed linear subspaces L ⊂ H such that J (L) = L⊥. The complex unitary group U (HJ), consisting of the elements of the orthogonal group of H which are complex linear for the given complex structure, acts transitively on Λ (H) and induces a natural linear connection in Λ (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 pL (=the orthogonal projection onto L) or symmetries ε{lunate}L = 2 pL - I, namely measuring tangent vectors with the operator norm. We show that for this metric the Hopf-Rinow theorem is valid in Λ (H): 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 ≤ ∞), with the Finsler metric given by the k-norm.