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: H₁→H₂ is Fredholm it is invertible modulo compact operators, i.e., if there exists a bounded linear operator g'\': H₂→H₁ 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.