Weak Symplectic Functional Analysis and General Spectral Flow Formula

We consider a continuous curve of self-adjoint Fredholm extensions of a curve of closed symmetric operators with fixed minimal domain $D_m$ and fixed {\it intermediate} domain $D_W$. Our main example is a family of symmetric generalized operators of Dirac type on a compact manifold with boundary with varying well-posed boundary conditions. Here $D_W$ is the first Sobolev space and $D_m$ the subspace of sections with support in the interior. We express the spectral flow of the operator curve by the Maslov index of a corresponding curve of Fredholm pairs of Lagrangian subspaces of the quotient Hilbert space $D_W/D_m$ which is equipped with continuously varying {\it weak symplectic structures} induced by the Green form. In this paper, we specify the continuity conditions; define the Maslov index in weak symplectic analysis; discuss the required weak inner Unique Continuation Property; derive a General Spectral Flow Formula; and check that the assumptions are natural and all are satisfied in geometric and pseudo-differential context. Applications are given to $L^2$ spectral flow formulae; to the splitting of the spectral flow on partitioned manifolds; and to linear Hamiltonian systems.