# Fredholm operator

Main page: Fredholm theory

In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator T : X → Y between two Banach spaces with finite-dimensional kernel $\displaystyle{ \ker T }$ and finite-dimensional (algebraic) cokernel $\displaystyle{ \operatorname{coker}T = Y/\operatorname{ran}T }$, and with closed range $\displaystyle{ \operatorname{ran}T }$. The last condition is actually redundant.[1]

The index of a Fredholm operator is the integer

$\displaystyle{ \operatorname{ind}T := \dim \ker T - \operatorname{codim}\operatorname{ran}T }$

or in other words,

$\displaystyle{ \operatorname{ind}T := \dim \ker T - \operatorname{dim}\operatorname{coker}T. }$

## Properties

Intuitively, Fredholm operators are those operators that are invertible "if finite-dimensional effects are ignored." The formally correct statement follows. A bounded operator T : X → Y between Banach spaces X and Y is Fredholm if and only if it is invertible modulo compact operators, i.e., if there exists a bounded linear operator

$\displaystyle{ S: Y\to X }$

such that

$\displaystyle{ \mathrm{Id}_X - ST \quad\text{and}\quad \mathrm{Id}_Y - TS }$

are compact operators on X and Y respectively.

If a Fredholm operator is modified slightly, it stays Fredholm and its index remains the same. Formally: 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, and the index is locally constant. More precisely, if T0 is Fredholm from X to Y, there exists ε > 0 such that every T in L(XY) with ||TT0|| < ε 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 $\displaystyle{ U \circ T }$ is Fredholm from X to Z and

$\displaystyle{ \operatorname{ind} (U \circ T) = \operatorname{ind}(U) + \operatorname{ind}(T). }$

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 unchanged under such a 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 U is Fredholm and T a strictly singular operator, then T + U is Fredholm with the same index.[2] The class of inessential operators, which properly contains the class of strictly singular operators, is the "perturbation class" for Fredholm operators. This means an operator $\displaystyle{ T\in B(X,Y) }$ is inessential if and only if T+U is Fredholm for every Fredholm operator $\displaystyle{ U\in B(X,Y) }$.

## Examples

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

$\displaystyle{ S(e_n) = e_{n+1}, \quad n \ge 0. \, }$

This operator S is injective (actually, isometric) and has a closed range of codimension 1, hence S is Fredholm with $\displaystyle{ \operatorname{ind}(S)=-1 }$. The powers $\displaystyle{ S^k }$, $\displaystyle{ k\geq0 }$, are Fredholm with index $\displaystyle{ -k }$. The adjoint S* is the left shift,

$\displaystyle{ S^*(e_0) = 0, \ \ S^*(e_n) = e_{n-1}, \quad n \ge 1. \, }$

The left shift S* is Fredholm with index 1.

If H is the classical Hardy space $\displaystyle{ H^2(\mathbf{T}) }$ on the unit circle T in the complex plane, then the shift operator with respect to the orthonormal basis of complex exponentials

$\displaystyle{ e_n : \mathrm{e}^{\mathrm{i} t} \in \mathbf{T} \mapsto \mathrm{e}^{\mathrm{i} n t}, \quad n \ge 0, \, }$

is the multiplication operator Mφ with the function $\displaystyle{ \varphi=e_1 }$. More generally, let φ be a complex continuous function on T that does not vanish on $\displaystyle{ \mathbf{T} }$, and let Tφ denote the Toeplitz operator with symbol φ, equal to multiplication by φ followed by the orthogonal projection $\displaystyle{ P:L^2(\mathbf{T})\to H^2(\mathbf{T}) }$:

$\displaystyle{ T_\varphi : f \in H^2(\mathrm{T}) \mapsto P(f \varphi) \in H^2(\mathrm{T}). \, }$

Then Tφ is a Fredholm operator on $\displaystyle{ H^2(\mathbf{T}) }$, with index related to the winding number around 0 of the closed path $\displaystyle{ t\in[0,2\pi]\mapsto \varphi(e^{it}) }$: the index of Tφ, as defined in this article, is the opposite of this winding number.

## Applications

Any 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.

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

The Atiyah-Jänich theorem identifies the K-theory K(X) of a compact topological space X with the set of homotopy classes of continuous maps from X to the space of Fredholm operators HH, where H is the separable Hilbert space and the set of these operators carries the operator norm.

## Generalizations

### Semi-Fredholm operators

A bounded linear operator T is called semi-Fredholm if its range is closed and at least one of $\displaystyle{ \ker T }$, $\displaystyle{ \operatorname{coker}T }$ is finite-dimensional. For a semi-Fredholm operator, the index is defined by

$\displaystyle{ \operatorname{ind}T=\begin{cases} +\infty,&\dim\ker T=\infty; \\ \dim\ker T-\dim\operatorname{coker}T,&\dim\ker T+\dim\operatorname{coker}T\lt \infty; \\ -\infty,&\dim\operatorname{coker}T=\infty. \end{cases} }$

### Unbounded operators

One may also define unbounded Fredholm operators. Let X and Y be two Banach spaces.

1. The closed linear operator $\displaystyle{ T:\,X\to Y }$ is called Fredholm if its domain $\displaystyle{ \mathfrak{D}(T) }$ is dense in $\displaystyle{ X }$, its range is closed, and both kernel and cokernel of T are finite-dimensional.
2. $\displaystyle{ T:\,X\to Y }$ is called semi-Fredholm if its domain $\displaystyle{ \mathfrak{D}(T) }$ is dense in $\displaystyle{ X }$, its range is closed, and either kernel or cokernel of T (or both) is finite-dimensional.

As it was noted above, the range of a closed operator is closed as long as the cokernel is finite-dimensional (Edmunds and Evans, Theorem I.3.2).

## Notes

1. Abramovich, Yuri A.; Aliprantis, Charalambos D. (2002). An Invitation to Operator Theory. Graduate Studies in Mathematics. 50. American Mathematical Society. p. 156. ISBN 978-0-8218-2146-6.
2. Kato, Tosio (1958). "Perturbation theory for the nullity deficiency and other quantities of linear operators". Journal d'Analyse Mathématique 6: 273—322. doi:10.1007/BF02790238.