**Memoirs of the American Mathematical Society**

2018;
123 pp;
Softcover

MSC: Primary 53;
Secondary 58

Print ISBN: 978-1-4704-2800-6

Product Code: MEMO/252/1201

List Price: $78.00

AMS Member Price: $46.80

MAA Member Price: $70.20

**Electronic ISBN: 978-1-4704-4371-9
Product Code: MEMO/252/1201.E**

List Price: $78.00

AMS Member Price: $46.80

MAA Member Price: $70.20

# The Maslov Index in Symplectic Banach Spaces

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*Bernhelm Booß-Bavnbek; Chaofeng Zhu*

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.

#### Table of Contents

# Table of Contents

## The Maslov Index in Symplectic Banach Spaces

- Cover Cover11
- Title page i2
- List of Figures vii8
- Preface ix10
- Introduction 112
- Part 1 . Maslov index in symplectic Banach spaces 920
- Part 2 . Applications in global analysis 5970
- Chapter 4. The desuspension spectral flow formula 6172
- 4.1. Short account of predecessor formulae 6172
- 4.2. Spectral flow for closed self-adjoint Fredholm relations 6677
- 4.3. Symplectic analysis of operators and relations 7182
- 4.4. Proof of the abstract spectral flow formula 7485
- 4.5. An application: A general desuspension formula for the spectral flow of families of elliptic boundary value problems 7687

- Backmatter 8293
- Appendix A. Perturbation of closed subspaces in Banach spaces 8394
- A.1. Some algebra facts 8495
- A.2. The gap topology 8495
- A.3. Continuity of operations of linear subspaces 8697
- A.4. Smooth family of closed subspaces in Banach spaces 92103
- A.5. Basic facts about symplectic Banach bundles 94105
- A.6. Embedding Banach spaces 95106
- A.7. Compact perturbations of closed subspaces 97108

- Bibliography 103114
- List of Symbols 109120
- Index of Names/Authors 113124
- Subject Index 115126

- Back Cover Back Cover1134