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The Maslov Index in Symplectic Banach Spaces


About this Title

Bernhelm Booß-Bavnbek and Chaofeng Zhu

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 252, Number 1201
ISBNs: 978-1-4704-2800-6 (print); 978-1-4704-4371-9 (online)
DOI: https://doi.org/10.1090/memo/1201
Published electronically: January 25, 2018
Keywords:Banach bundles, Calderón projection, Cauchy data spaces, elliptic operators, Fredholm pairs, desuspension spectral flow formula, Lagrangian subspaces, Maslov index, symplectic reduction, unique continuation property, variational properties, weak symplectic structure, well-posed boundary conditions

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Table of Contents


Chapters

  • Preface
  • Introduction

Part 1. Maslov index in symplectic Banach spaces

  • Chapter 1. General theory of symplectic analysis in Banach spaces
  • Chapter 2. The Maslov index in strong symplectic Hilbert space
  • Chapter 3. The Maslov index in Banach bundles over a closed interval

Part 2. Applications in global analysis

  • Chapter 4. The desuspension spectral flow formula
  • Appendix A. Perturbation of closed subspaces in Banach spaces

Abstract


We consider a curve of [[subindex]]Fredholm pair!of Lagrangian subspaces!curveFredholm pairs of [[subindex]]Lagrangian subspacesLagrangian subspaces in a fixed [[subindex]]Symplectic form!weak [[subindex]]Banach space!with varying weak symplectic structureBanach 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 Maslov index of the curve 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 [[subindex]]Symplectic reduction symplectic reduction, while recovering all the standard properties of the Maslov index.As an application, we consider curves of elliptic operators which have varying [[subindex]]Principal symbolprincipal symbol, varying [[subindex]]Domain!maximalmaximal domain and are [[subindex]]Elliptic differential operators!of not-Dirac typenot 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 formula of the spectral flow on [[subindex]]Partitioned manifoldpartitioned manifolds.

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