Fredholm index

The previous post on kernels and cokernels mentioned that for a linear operator T:VW, the index ofT is defined as the difference between the dimension of its kernel and the dimension of its cokernel:

index T = dim ker T- dim cokerT.

The index was first called the Fredholm index, because of it came up in Fredholm's investigation of integral equations. (More on this work in the next post.)

Robustness

The index of a linear operator is robust in the following sense. IfV andW are Banach spaces and T:VW is a continuous linear operator, then there is an open set around T in the space of continuous operators fromV toW on which the index is constant. In other words, small changes toT don't change its index.

Small changes toT may alter the dimension of the kernel or the dimension of the cokernel, but they don't alter their difference.

Relation to Fredholm alternative

The next post discusses the Fredholm alternative theorem. It says that ifK is a compact linear operator on a Hilbert space andI is the identity operator, then the Fredholm index of I - K is zero. The post will explain how this relates to solving linear (integral) equations.

Analogy to Euler characteristic

We can make an exact sequence with the spacesV andW and the kernel and cokernel ofT as follows:

0 ker T V W coker T 0

All this means is that the image of one map is the kernel of the next.

We can take the alternating sum of the dimensions of the spaces in this sequence:

dim kerT - dimV + dimW - dim cokerT.

IfV andW have the same finite dimension, then this alternating sum equals the index ofT.

The Euler characteristic is also an alternating sum. For a simplex, the Euler characteristic is defined by

V -E +F

whereV is the number of vertices,E the number of edges, andF the number of faces. We can extend this to higher dimensions as the number of zero-dimensional object (vertices), minus the number of one-dimensional objects (edges), plus the number of two-dimensional objects, minus the number of three dimensional objects,etc.

A more sophisticated definition of Euler characteristic is the alternating sum of the dimensions of cohomology spaces. These also form an exact sequence.

The Atiyah-Singer index theorem says that for elliptic operators on manifolds, two kinds of index are equal: the analytical index and the topological index. The analytical index is essentially the Fredholm index. The topological index is derived from topological information about the manifold.

This is analogous to the Gauss-Bonnet theorem that says you can find the Euler characteristic, a topological invariant, by integrating Gauss curvature, an analytic calculation.

Other posts in this series

This is the middle post in a series of three. The first was on kernels and cokernels, and the next is on the Fredholm alternative.

The post Fredholm index first appeared on John D. Cook.