Self-Adjoint Fredholm Operators And Spectral Flow

Abstract

We study the topology of the nontrivial component, , of self-adjoint Fredholm operators on a separable Hilbert space. In particular, if {B t} is a path of such operators, we can associate to {B t} an integer, sf({B t}), called the spectral flow of the path. This notion, due to M. Atiyah and G. Lusztig, assigns to the path {B t} the net number of eigenvalues (counted with multiplicities) which pass through 0 in the positive direction. There are difficulties in making this precise — the usual argument involves looking at the graph of the spectrum of the family (after a suitable perturbation) and then counting intersection numbers with y = 0. We present a completely different approach using the functional calculus to obtain continuous paths of eigenprojections (at least locally) of the form . The spectral flow is then defined as the dimension of the nonnegative eigenspace at the end of this path minus the dimension of the nonnegative eigenspace at the beginning. This leads to an easy proof that spectral flow is a well-defined homomorphism from the homotopy groupoid of onto Z. For the sake of completeness we also outline the seldom-mentioned proof that the restriction of spectral flow to is an isomorphism onto Z.