## Abstract

The authors consider a curve of Fredholm pairs of Lagrangian subspaces in a fixed Banach space with continuously varying weak symplectic structures. Assuming vanishing index, they obtain intrinsically a continuously varying splitting of the total Banach space into pairs of symplectic subspaces. Using such decompositions the authors define the Maslov index of the curve by symplectic reduction to the classical finite-dimensional case. The authors 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, the authors 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, the authors derive a desuspension spectral flow formula for varying well-posed boundary conditions on manifolds with boundary and obtain the splitting formula of the spectral flow on partitioned manifolds.

As an application, the authors 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, the authors derive a desuspension spectral flow formula for varying well-posed boundary conditions on manifolds with boundary and obtain the splitting formula of the spectral flow on partitioned manifolds.

Original language | English |
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Book series | Memoirs of the American Mathematical Society |

Volume | 252 |

Issue number | 1201 |

Pages (from-to) | 1-134 |

Number of pages | 123 |

ISSN | 0065-9266 |

DOIs | |

Publication status | Published - 2 Apr 2018 |

### Bibliographical note

MSC: Primary 53; Secondary 58## Keywords

- Banach bundles
- Calder´on projection
- Cauchy data spaces
- Fredholm pairs
- Lagrangian subspaces
- Maslov index
- desuspension spectral flow formula
- elliptic operators
- symplectic reduction
- unique continuation property
- variational properties
- weak symplectic structur
- well-posed boundary conditions