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The Maslov Index in Symplectic Banach Spaces
About this Title
Bernhelm Booß-Bavnbek, Department of Sciences, Systems and Models/IMFUFA, Roskilde University, 4000 Roskilde, Denmark and Chaofeng Zhu, Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
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
MSC: Primary 53D12; Secondary 58J30
Table of Contents
Chapters
- Preface
- Introduction
1. Maslov index in symplectic Banach spaces
- 1. General theory of symplectic analysis in Banach spaces
- 2. The Maslov index in strong symplectic Hilbert space
- 3. The Maslov index in Banach bundles over a closed interval
2. Applications in global analysis
- 4. The desuspension spectral flow formula
- A. Perturbation of closed subspaces in Banach spaces
Abstract
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 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 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 formula of the spectral flow on partitioned manifolds.
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