# Fredholm operator

A

**Fredholm operator** is a

bounded linear operator between two Hilbert spaces whose

range is closed and whose

kernel and

cokernel are finite-dimensional. Equivalently, an operator

*f*:

*H*_{1}→

*H*_{2} is Fredholm it is invertible modulo compact operators, i.e., if there exists a bounded linear operator

*g'\': *H

_{2}→H

_{1} such that Id_{}H

_{1} - gf

* and Id*_{}H

_{2} - fg

* are compact operators on *H

_{1} and H''

_{2} respectively.

A Fredholm operator has a well-defined index, which remains constant under continuous deformation of the operator itself. An elliptic differential operator can be extended to a Fredholm operator. The Atiyah-Singer index theorem gives a topological characterization of the index.