  We consider a curve of Fredholm pairs of Lagrangian subspaces in a fixed
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 define the curve's Maslov index by symplectic reduction to
the classical finite-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 flow formula for varying well-posed boundary conditions on manifolds
with boundary and obtain the splitting of the spectral flow on partitioned
manifolds.

Symplectic Geometry;Differential Geometry;Functional Analysis