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

$${\mathrm{ind}}_{\mathrm{an}}(E_p,D_p)=\sum _p(-1{)}^{+}\dim \ker {\Delta }_p.$$ind_{an}(E_p, D_p) = \sum_p (-1)^+ dim ker \Delta_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