The Maslov index and the spectral flow—revisited

Abstract

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 selfadjoint first-order operators. We particularly pay attention to the continuity of the latter path of operators, where we consider the gap-metric on the set of all closed operators on a Hilbert space. Finally, we obtain from Cappell, Lee and Miller's theorem a spectral flow formula for linear Hamiltonian systems which generalises a recent result of Hu and Portaluri.

On a Comparison Principle and the Uniqueness of Spectral Flow

“…)), which is negative definite by assumption. It follows from (11) that sf(h(•, 1)) ≤ 0, and so the theorem is shown.…”

“…)), which is positive definite by assumption. Hence it follows from (11) that sf(h(•, 0)) ≥ 0. Let us now consider the other case and assume that s * is a crossing of h(•, 1).…”

“…Let us first note that h(•, 0) and h(•, 1) are both differentiable paths of selfadjoint Fredholm operators as above, where the spaces W are D(A 0 ) and D(A 1 ) with the graph norms of these operators. Therefore we can use (11) to compute their spectral flows. Let us consider h(•, 0) first.…”

“…As this invariant has been studied in numerous references, we will not provide further details and refer to [6] and [11]. Let us note, however, that there are different non-equivalent approaches to the Maslov index and here we will always consider µ Mas (Λ 1 , Λ 2 ) as constructed in the previous references.…”

“…It is well known that A λ are selfadjoint Fredholm operators (see, e.g., [6]). Moreover, the path A = {A λ } λ∈I is gap-continuous by [11,Thm. 1.1].…”