Lagrangian Grassmannian in infinite dimension

ESTEBAN ANDRUCHOW; GABRIEL LAROTONDA

JOURNAL OF GEOMETRY AND PHYSICS

North-Holland

Lugar: Amsterdam, Holanda; Año: 2009 vol. 59 p. 306 - 320

0393-0440

Given a complex structure J on a real (finite or infinite dimensional) Hilbert space $h$, we study the geometry of the Lagrangian Grassmannian L(H) of H, i.e. the set of closed linear subspaces L subset H such that J(L)=L^perp. 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 L(H) and induces a natural linear connection there. 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 e_L=2p_L-I, namely measuring tangent vectors with the operator norm. We show that for this metric the Hopf-Rinow theorem is valid in L(H): a geodesic joining a pair of Lagrangian subspaces can be chosen to be of minimal length. We extend these results to the classical Banach-Lie groups of Schatten.