The Index and Other Properties of Elliptic Operators

Abstract

We now enumerate, mostly without proof, the most important properties of elliptic operators. Let A be an elliptic operator on a compact manifold M, that is, an elliptic operator on the space of sections Γ = Γ(ξ) of a vector bundle ξ = (E, M, R n , p) over M. The kernel of A, denoted Ker A, is the set of solutions of the equation Af = 0; the dimension l(a) of Ker A is also called the number of zero modes of A. It can be proved that all eigenvalues of A have finite multiplicity, and in particular that l(A) is finite. It can also be shown that the image of A, that is, the set of g ∈ Γ such that g = Af for some f ∈ G, is a subspace of finite codimension r(A) in Γ; in other words, that set is the space of solutions of a finite set of linear equations.