We prove that any countable family of Lagrangian subspaces of a symplectic Hilbert space admits a common complementary Lagrangian. The proof of this puzzling result, which is not totally elementary also in the finite dimensional case, is obtained as an application of the spectral theorem for unbounded self-adjoint operators.
symplectic Hilbert spaces; Lagrangian subspaces; Lagrangian Grassmannian; unbounded self-adjoint operators; spectral theorem
