On a Comparison Principle and the Uniqueness of Spectral Flow

Abstract

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, we consider homotopy invariance properties of the spectral flow and establish a simple formula which comprises its classical homotopy invariance and yields a comparison theorem for the spectral flow under compact perturbations. We apply our result to the existence of non-trivial solutions of boundary value problems of Hamiltonian systems. Secondly, the spectral flow was axiomatically characterised by Lesch, and by Ciriza, Fitzpatrick and Pejsachowicz under the assumption that the endpoints of the paths of selfadjoint Fredholm operators are invertible. We propose a different approach to the uniqueness of spectral flow which lifts this additional assumption. As application of the latter result, we discuss the relation between the spectral flow and the Maslov index in symplectic Hilbert spaces.

Dynamical convexity and closed orbits on symmetric spheres

“…Each L s z,A i is a self-adjoint Fredholm operator. Therefore, we have, for a fixed z and A i , the corresponding spectral flow sfpL s z,A i q P Z, see, for instance, [4,20,22,23] and references therein. The z-index of Γ C i , as defined in [21, Definition 2.3] and [20,Definition 2.8], is given by…”

The Equivariant Spectral Flow and Bifurcation of Periodic Solutions of Hamiltonian Systems

“…, N . Moreover, the spectral flow can be uniquely characterised by some of its properties (see, e.g., [19], [9], [27]), and among them is its quite remarkable homotopy invariance. We do not recall here any of these properties, as most of them will follow as special cases of our equivariant spectral flow, which we discuss in the next section.…”

“…Our final aim of this section is to show the homotopy invariance of the G-equivariant spectral flow. Here we follow the approach of [27], which is based on [22], and begin by the following proposition.…”

“…Finally, let us point out that further homotopy invariance properties can be obtained from Proposition 2.8, e.g., the invariance under free homotopies of loops in F S G (W, H) (cf. [27]).…”

“…We now see from (26) that i.e., F 1 and F 2 are families of G-equivariant maps. Note that, as X and Y are invariant, g(x, y) = (gx, gy) for g ∈ G and (x, y) ∈ X ⊕ Y = H. As the lemma holds in the non-equivariant case, there is a unique family of C 1 -maps η : I × B X → Y that satisfies (27). Now B X is invariant as G acts orthogonally, and thus F 2 (λ, gx, η(λ, gx)) = 0, g ∈ G, x ∈ B X F 2 (λ, gx, gη(λ, x)) = gF 2 (λ, x, η(λ, x)) = 0, g ∈ G, x ∈ B X .…”