The Maslov Index in Symplectic Banach Spaces.

Cover -- Title page -- List of Figures -- Preface -- Introduction -- Part 1 . Maslov index in symplectic Banach spaces -- Chapter 1. General theory of symplectic analysis in Banach spaces -- 1.1. Dual pairs and double annihilators -- 1.2. Basic symplectic concepts -- 1.3. Natural decomposition of induced by a Fredholm pair of Lagrangian subspaces with vanishing index -- 1.4. Symplectic reduction of Fredholm pairs -- Chapter 2. The Maslov index in strong symplectic Hilbert space -- 2.1. The Maslov index via unitary generators -- 2.2. The Maslov index in finite dimensions -- 2.3. Properties of the Maslov index in strong symplectic Hilbert space -- Chapter 3. The Maslov index in Banach bundles over a closed interval -- 3.1. The Maslov index by symplectic reduction to a finite-dimensional subspace -- 3.2. Calculation of the Maslov index -- 3.3. Invariance of the Maslov index under symplectic operations -- 3.4. The Hörmander index -- Part 2 . Applications in global analysis -- Chapter 4. The desuspension spectral flow formula -- 4.1. Short account of predecessor formulae -- 4.2. Spectral flow for closed self-adjoint Fredholm relations -- 4.3. Symplectic analysis of operators and relations -- 4.4. Proof of the abstract spectral flow formula -- 4.5. An application: A general desuspension formula for the spectral flow of families of elliptic boundary value problems -- Backmatter -- Appendix A. Perturbation of closed subspaces in Banach spaces -- A.1. Some algebra facts -- A.2. The gap topology -- A.3. Continuity of operations of linear subspaces -- A.4. Smooth family of closed subspaces in Banach spaces -- A.5. Basic facts about symplectic Banach bundles -- A.6. Embedding Banach spaces -- A.7. Compact perturbations of closed subspaces -- Bibliography -- List of Symbols -- Index of Names/Authors -- Subject Index -- Back Cover.

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.

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