The Hilbert-Schmidt Lagrangian Grassmannian in infinite dimension

In this paper we study the action of the symplectic operators which are a perturbation of the identity by a Hilbert-Schmidt operator in the Lagrangian Grassmannian manifold. We prove several geometric properties using the quotient norm in the tangent spaces. We determinate the geodesic curves of this structure and we study the completeness of the corresponding geodesic distance.