# 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, with finite-dimensional kernel and cokernel, and with closed range. (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

$S:Y\to X$ such that

$\mathrm {Id} _{X}-ST\quad {\text{and}}\quad \mathrm {Id} _{Y}-TS$ are compact operators on X and Y respectively.

The index of a Fredholm operator is

$\mathrm {ind} \,T:=\dim \ker T-\mathrm {codim} \,\mathrm {ran} \,T$ or in other words,

$\mathrm {ind} \,T:=\dim \ker T-\mathrm {dim} \,\mathrm {coker} \,T;$ see dimension, kernel, codimension, range, and cokernel.

## Properties

The set of Fredholm operators from X to Y is open in the Banach space L(XY) of bounded linear operators, equipped with the operator norm. More precisely, when T0 is Fredholm from X to Y, there exists ε > 0 such that every T in L(XY) with Template:Nowrap begin||TT0|| < εTemplate:Nowrap end is Fredholm, with the same index as that of T0.

When T is Fredholm from X to Y and U Fredholm from Y to Z, then the composition $U\circ T$ is Fredholm from X to Z and

$\mathrm {ind} (U\circ T)=\mathrm {ind} (U)+\mathrm {ind} (T).$ When T is Fredholm, the transpose (or adjoint) operator T ′ is Fredholm from Y ′ to X ′, and ind(T ′) = −ind(T). When X and Y are Hilbert spaces, the same conclusion holds for the Hermitian adjoint T.

When T is Fredholm and K a compact operator, then T + K is Fredholm. The index of T remains constant under compact perturbations of T. This follows from the fact that the index i(s) of T + sK is an integer defined for every s in [0, 1], and i(s) is locally constant, hence i(1) = i(0).

Invariance by perturbation is true for larger classes than the class of compact operators. For example, when T is Fredholm and S a strictly singular operator, then T + S is Fredholm with the same index. A bounded linear operator S from X to Y is strictly singular when its restriction to any infinite dimensional subspace X0 of X fails to be an into isomorphism, that is:

$\inf\{\|Sx\|:x\in X_{0},\,\|x\|=1\}=0.\,$ ## Examples

Let H be a Hilbert space with an orthonormal basis {en} indexed by the non negative integers. The (right) shift operator S on H is defined by

$S(e_{n})=e_{n+1},\quad n\geq 0.\,$ This operator S is injective (actually, isometric) and has a closed range of codimension 1, hence S is Fredholm with ind(S) = −1. The powers Sk, k ≥ 0, are Fredholm with index −k. The adjoint S is the left shift,

$S^{*}(e_{0})=0,\ \ S^{*}(e_{n})=e_{n-1},\quad n\geq 1.\,$ The left shift S is Fredholm with index 1.

If H is the classical Hardy space H2(T) on the unit circle T in the complex plane, then the shift operator with respect to the orthonormal basis of complex exponentials

$e_{n}:\mathrm {e} ^{\mathrm {i} t}\in \mathbf {T} \rightarrow \mathrm {e} ^{\mathrm {i} nt},\quad n\geq 0,\,$ is the multiplication operator Mφ with the function φ = e1. More generally, let φ be a complex continuous function on T that does not vanish on T, and let Tφ denote the Toeplitz operator with symbol φ, equal to multiplication by φ followed by the orthogonal projection P from L2(T) onto H2(T):

$T_{\varphi }:f\in H^{2}(\mathrm {T} )\rightarrow P(f\varphi )\in H^{2}(\mathrm {T} ).\,$ Then Tφ is a Fredholm operator on H2(T), with index related to the winding number around 0 of the closed path t ∈ [0, 2 π] → φ(e i t) : the index of Tφ, as defined in this article, is the opposite of this winding number.

## Applications

The Atiyah-Singer index theorem gives a topological characterization of the index of certain operators on manifolds.

An elliptic operator can be extended to a Fredholm operator. The use of Fredholm operators in partial differential equations is an abstract form of the parametrix method.