Fredholm operator

A Fredholm operator is 'invertible up to a finite-dimensional error'. It might fail to be injective and fail to be surjective, but only in finitely many dimensions on each side — a finite-dimensional kernel and a finite-dimensional cokernel. The bookkeeping of how badly it fails on each side is captured by a single integer, the index, which is remarkably robust.

A bounded operator T between Banach spaces is Fredholm if its kernel ker T is finite-dimensional, its range is closed, and the cokernel (the quotient of the target by the range) is finite-dimensional. Its index is the integer ind(T) = dim(ker T) - dim(coker T). The index is invariant under small perturbations and under perturbation by any compact operator, and it is additive under composition: ind(ST) = ind(S) + ind(T).

Fredholm operators are exactly the operators that become invertible in the Calkin algebra (bounded operators modulo compact operators) — this is Atkinson's theorem. Their stability is the reason the index is the basic object in index theory, linking analysis to topology (the Atiyah–Singer index theorem). For a square matrix in finite dimensions the rank–nullity theorem forces the index to be 0; nonzero index is a genuinely infinite-dimensional phenomenon, exhibited by the shift operators.