On the Fredholm Lagrangian Grassmannian, spectral flow and ODEs in Hilbert spaces

  title={On the Fredholm Lagrangian Grassmannian, spectral flow and ODEs in Hilbert spaces},
  author={Nils Waterstraat},
  journal={Journal of Differential Equations},
  • Nils Waterstraat
  • Published 3 March 2018
  • Mathematics
  • Journal of Differential Equations
1 Citations
On a comparison principle and the uniqueness of spectral flow
The spectral flow is a well‐known quantity in spectral theory that measures the variation of spectra about 0 along paths of selfadjoint Fredholm operators. The aim of this work is twofold. Firstly,


The Maslov Index in Symplectic Banach Spaces
We consider a curve of Fredholm pairs of Lagrangian subspaces in a xed Banach space with continuously varying weak symplectic structures. Assuming vanishing index, we obtain intrin- sically a
Ordinary differential operators in Hilbert spaces and Fredholm pairs
Abstract. Let $A(t)$ be a path of bounded operators on a real Hilbert space, hyperbolic at $\pm \infty$. We study the Fredholm theory of the operator $F_A=d/dt-A(t)$. We relate the Fredholm property
The uniqueness of the spectral flow on spaces of unbounded self--adjoint Fredholm operators
We discuss several natural metrics on spaces of unbounded self--adjoint operators and their relations, among them the Riesz and the graph metric. We show that the topologies of the spaces of Fredholm
The Spectral Flow and the Maslov Index
exist and have no zero eigenvalue. A typical example for A(t) is the div-grad-curl operator on a 3-manifold twisted by a connection which depends on t. Atiyah et al proved that the Fredholm index of
The Index Bundle for Gap-Continuous Families, Morse-Type Index Theorems and Bifurcation
We generalise the well known index bundle construction of Atiyah and Janich to families of (generally unbounded) Fredholm operators which are assumed to be continuous with respect to the gap
The Maslov index and the spectral flow—revisited
We give an elementary proof of a celebrated theorem of Cappell, Lee and Miller which relates the Maslov index of a pair of paths of Lagrangian subspaces to the spectral flow of an associated path of
Self-Adjoint Fredholm Operators And Spectral Flow
  • J. Phillips
  • Mathematics
    Canadian Mathematical Bulletin
  • 1996
Abstract We study the topology of the nontrivial component, , of self-adjoint Fredholm operators on a separable Hilbert space. In particular, if {Bt } is a path of such operators, we can associate to
Unbounded Fredholm Operators and Spectral Flow
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
Index theory for heteroclinic orbits of Hamiltonian systems
Index theory revealed its outstanding role in the study of periodic orbits of Hamiltonian systems and the dynamical consequences of this theory are enormous. Although the index theory in the periodic
Instanton Floer homology with Lagrangian boundary conditions
In this paper we define instanton Floer homology groups for a pair consisting of a compact oriented 3–manifold with boundary and a Lagrangian submanifold of the moduli space of flat SU.2/–connections