A formula of local Maslov index and applications

In this paper, we explicitly express the local Maslov index by a Maslov index in finite dimensional case without symplectic reduction. Then we calculate the Maslov index for the path of pairs of Lagrangian subspaces in triangular form. In particular, we get the Maslov-type index of a given symplectic path in triangle form. As applications, we calculate the splitting numbers of the symplectic matrix in triangle form, dependence of iteration theory on triangular frames and mod 2 Maslov-type index for a real symplectic path. We study the continuity of families of bounded linear relations and families of bounded linear operators acting on closed linear subspaces as technique preparations.