The Maslov index in symplectic Banach spaces

We consider a curve of Fredholm pairs of Lagrangian subspaces in a xed
Banach space with continuously varying weak symplectic structures. Assuming
vanishing index, we obtain intrinsically a continuously varying splitting of the total Banach space into pairs of symplectic subspaces. Using such decompositions we dene the Maslov index of the curve by symplectic reduction to the classical nite-dimensional case. We prove the transitivity of repeated symplectic reductions and obtain the invariance of the Maslov index under symplectic reduction, while recovering all the standard properties of the Maslov index.
As an application, we consider curves of elliptic operators which have varying
principal symbol, varying maximal domain and are not necessarily of Dirac type.
For this class of operator curves, we derive a desuspension spectral ow formula for varying well-posed boundary conditions on manifolds with boundary and obtain the splitting of the spectral ow on partitioned manifolds.