Main Page | See live article | Alphabetical index

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: H1H2 is Fredholm it is invertible modulo compact operators, i.e., if there exists a bounded linear operator g'\': H2H1 such that IdH1 - gf and IdH2 - fg are compact operators on H1 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.