nLab analytical index

Idea

By pseudo-differential analysis an elliptic operator acting on sections of two vector bundles on a manifold is a Fredholm operator and hence has closed kernel and cokernel of finite dimension. The difference of these two dimensions is the analytical index of the operator.

More generally, for $(E_p, D_p)$ an elliptic complex, its analytical index is the alternating sum

$$ind_{an}(E_p, D_p) = \sum_p (-1)^p dim (ker (D_p)) \,.$$

Properties

This index does not the depend of the Sobolev space used to get a bounded operator (by elliptic regularity the kernel is made up of smooth sections and the same for the cokernel as it is the kernel of the adjoint). By topological K-theory one can associate to it also a topological index. The Atiyah-Singer index theorem say that this two indexes coincide.

Related concepts

• index theory

  • topological index

  • analytical index

  • Atiyah-Singer index theorem