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Fredholm operator

## Summary

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 ${\displaystyle \mathrm {coker} \,T=Y/\mathrm {ran} \,T}$, and with closed range ${\displaystyle \mathrm {ran} \,T}$. The last condition is actually redundant.[1]

The index of a Fredholm operator is the integer

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

or in other words,

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

## 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

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

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 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

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

This operator S is injective (actually, isometric) and has a closed range of codimension 1, hence S is Fredholm with ${\displaystyle \mathrm {ind} (S)=-1}$ . The powers ${\displaystyle S^{k}}$ , ${\displaystyle k\geq 0}$ , are Fredholm with index ${\displaystyle -k}$ . The adjoint S* is the left shift,

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

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

${\displaystyle e_{n}:\mathrm {e} ^{\mathrm {i} t}\in \mathbf {T} \rightarrow \mathrm {e} ^{\mathrm {i} nt},\quad n\geq 0,\,}$

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 ${\displaystyle P:L^{2}(\mathbf {T} )\to H^{2}(\mathbf {T} )}$ :

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

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

### B-Fredholm operators

For each integer ${\displaystyle n}$ , define ${\displaystyle T_{n}}$  to be the restriction of ${\displaystyle T}$  to ${\displaystyle R(T^{n})}$  viewed as a map from ${\displaystyle R(T^{n})}$  into ${\displaystyle R(T^{n})}$  ( in particular ${\displaystyle T_{0}=T}$ ). If for some integer ${\displaystyle n}$  the space ${\displaystyle R(T^{n})}$  is closed and ${\displaystyle T_{n}}$  is a Fredholm operator, then ${\displaystyle T}$  is called a B-Fredholm operator. The index of a B-Fredholm operator ${\displaystyle T}$  is defined as the index of the Fredholm operator ${\displaystyle T_{n}}$ . It is shown that the index is independent of the integer ${\displaystyle n}$ . B-Fredholm operators were introduced by M. Berkani in 1999 as a generalization of Fredholm operators.[3]

### Semi-Fredholm operators

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

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

### 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. ^ Yuri A. Abramovich and Charalambos D. Aliprantis, "An Invitation to Operator Theory", p.156
2. ^ T. Kato, "Perturbation theory for the nullity deficiency and other quantities of linear operators", J. d'Analyse Math. 6 (1958), 273–322.
3. ^ Berkani Mohammed: On a class of quasi-Fredholm operators. Integral Equations and Operator Theory, 34, 2 (1999), 244-249 [1]

## References

• D.E. Edmunds and W.D. Evans (1987), Spectral theory and differential operators, Oxford University Press. ISBN 0-19-853542-2.
• A. G. Ramm, "A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators", American Mathematical Monthly, 108 (2001) p. 855 (NB: In this paper the word "Fredholm operator" refers to "Fredholm operator of index 0").
• Weisstein, Eric W. "Fredholm's Theorem". MathWorld.
• B.V. Khvedelidze (2001) [1994], "Fredholm theorems", Encyclopedia of Mathematics, EMS Press
• Bruce K. Driver, "Compact and Fredholm Operators and the Spectral Theorem", Analysis Tools with Applications, Chapter 35, pp. 579–600.
• Robert C. McOwen, "Fredholm theory of partial differential equations on complete Riemannian manifolds", Pacific J. Math. 87, no. 1 (1980), 169–185.
• Tomasz Mrowka, A Brief Introduction to Linear Analysis: Fredholm Operators, Geometry of Manifolds, Fall 2004 (Massachusetts Institute of Technology: MIT OpenCouseWare)