Fredholm-Lagrangian-Grassmannian and the Maslov index
Kenro Furutani
TL;DR
The article develops a unified, functional-analytic framework for the Fredholm-Lagrangian-Grassmannian and the Maslov index, using the Souriau map to connect Lagrangian subspaces with unitary operators and proving that π1 of the Fredholm-Lagrangian-Grassmannian is ℤ. It defines the infinite-dimensional Maslov index through spectral flow-like constructions, introduces Hörmander and complex Kashiwara indices, and establishes a universal covering and Maslov line bundle; a central reduction theorem connects infinite-dimensional data to finite-dimensional reductions via polarization. The framework is then applied to closed symmetric operators and Cauchy data spaces, linking the Maslov index to spectral flow and providing an operator-theoretic bridge to K-theory. These results yield a coherent, elementary approach to Maslov-type invariants in infinite dimensions with broad implications for elliptic and Dirac-type operators and boundary value problems.
Abstract
We explain the topology of the space, so called, Fredholm-Lagrangian-Grassmannain and the quantity ``Maslov index'' for paths in this space based on the standard theory of Functional Analysis. Our standing point is to define the Maslov index for arbitrary paths in terms of the fundamental spectral property of the Fredholm operators, which was first recognized by J. Phillips and used to define the ``Spectral flow''. We tried to make the arguments to be all elementary and we summarize basic facts for this article from Functional Analysis in the Appendix.