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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 H2 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.
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