Fredholm Operator

In mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations. It is named in honour of Erik Ivar Fredholm.

A Fredholm operator is a bounded linear operator between two Banach spaces whose kernel and cokernel are finite-dimensional and whose range is closed. (The last condition is actually redundant.) Equivalently, an operator T : X → Y is Fredholm if it is invertible modulo compact operators, i.e., if there exists a bounded linear operator

such that

are compact operators on X and Y respectively.

The index of a Fredholm operator is

or equivalently,

see dimension, kernel, codimension, range, and cokernel.