Stabilities in Banach spaces with skew-adjoint Fredholm operators

In this paper, we prove the stability theorems for the isotropic perturbations of Lagrangian subspaces and maximal isotropic subspaces in strong symplectic Banach spaces. Then we prove a stability theorem for the mod $2$ dimensions of kernel of skew-adjoint Fredholm linear relations between real Banach spaces with index $0$. Finally we gives the path components of the set of skew-adjoint Fredholm linear relations between real Banach spaces with indices $0$.