Quantum Field Theory and Topology pp 163-168 | Cite as

# 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.

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