# The Maslov index in weak symplectic functional analysis

@article{BoossBavnbek2013TheMI, title={The Maslov index in weak symplectic functional analysis}, author={Bernhelm Booss-Bavnbek and Chaofeng Zhu}, journal={Annals of Global Analysis and Geometry}, year={2013}, volume={44}, pages={283-318} }

We recall the Chernoff-Marsden definition of weak symplectic structure and give a rigorous treatment of the functional analysis and geometry of weak symplectic Banach spaces. We define the Maslov index of a continuous path of Fredholm pairs of Lagrangian subspaces in continuously varying symplectic Banach spaces. We derive basic properties of this Maslov index and emphasize the new features appearing.

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#### References

SHOWING 1-10 OF 72 REFERENCES

The Maslov index and a generalized Morse index theorem for non-positive definite metrics

- Mathematics
- 2000

Abstract We present an extension of the celebrated Morse index theorem in Riemannian geometry to the case of geodesics in pseudo-Riemannian manifolds. It is considered the case that both endpoints… Expand

A K-Theoretic Proof of the Morse Index Theorem in Semi-Riemannian Geometry

- Mathematics
- 2010

We give a short proof of the Morse index theorem for geodesics in semi-Riemannian manifolds by using K-theory. This makes the Morse index theorem reminiscent of the Atiyah-Singer index theorem for… Expand

A relative morse index for the symplectic action

- Mathematics
- 1988

The notion of a Morse index of a function on a finite-dimensional manifold cannot be generalized directly to the symplectic action function a on the loop space of a manifold. In this paper, we define… Expand

MASLOV-TYPE INDEX THEORY FOR SYMPLECTIC PATHS AND SPECTRAL FLOW （II）

- Mathematics
- 1999

Based on the spectral flow and the stratification structures of the symplectic group Sp(2n, C),the Maslov-type index theory and its generalization, the ω-index theory parameterized by all ω on the… Expand

Unbounded Fredholm Operators and Spectral Flow

- Mathematics
- Canadian Journal of Mathematics
- 2005

Abstract We study the gap (= “projection norm” = “graph distance”) topology of the space of all (not necessarily bounded) self-adjoint Fredholm operators in a separable Hilbert space by the Cayley… Expand

Maslov index in the infinite dimension and a splitting formula for a spectral flow

- Mathematics
- 2002

First, we prove a local spectral flow formula (Theorem 3.7) for a differentiable curve of selfadjoint Fredholm operators. This formula enables us to prove in a simple way a general spectral flow… Expand

Weak Symplectic Functional Analysis and General Spectral Flow Formula

- Mathematics
- 2004

We consider a continuous curve of self-adjoint Fredholm extensions of a curve of closed symmetric operators with fixed minimal domain $D_m$ and fixed {\it intermediate} domain $D_W$. Our main example… Expand

A Morse index theorem for perturbed geodesics on semi-Riemannian manifolds

- Mathematics
- 2005

Perturbed geodesics are trajectories of particles moving on
a semi-Riemannian manifold in the presence of a potential. Our
purpose here is to extend to perturbed geodesics on
semi-Riemannian… Expand

Elliptic Boundary Problems for Dirac Operators

- Mathematics
- 1993

The major goal of this book is to make the theory of elliptic boundary problems accessible to mathematicians and physicists working in global analysis and operator algebras. The book is about… Expand

The Morse index theorem for regular Lagrangian systems

- Mathematics
- 2003

In this paper, we prove a Morse index theorem for the index form of regular Lagrangian system with selfadjoint boundary condition.