# 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.^{[1]}) 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 in other words,

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

## Properties

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

When *T* is Fredholm from *X* to *Y* and *U* Fredholm from *Y* to *Z*, then the composition is Fredholm from *X* to *Z* and

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* + *s* *K* 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.^{[2]} A bounded linear operator *S* from *X* to *Y* is **strictly singular** when its restriction to any infinite dimensional subspace *X*_{0} of *X* fails to be an into isomorphism, that is:

## Examples

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

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 *S*^{k}, *k* ≥ 0, are Fredholm with index −*k*. The adjoint *S*^{∗} is the left shift,

The left shift *S*^{∗} is Fredholm with index 1.

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

is the multiplication operator *M*_{φ} with the function *φ* = *e*_{1}. More generally, let *φ* be a complex continuous function on **T** that does not vanish on **T**, and let *T*_{φ} denote the Toeplitz operator with symbol *φ*, equal to multiplication by *φ* followed by the orthogonal projection *P* from *L*^{2}(**T**) onto *H*^{2}(**T**):

Then *T*_{φ} is a Fredholm operator on *H*^{2}(**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 *T*_{φ}, 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.

## B-Fredholm operators

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

## Notes

- ↑ Yuri A. Abramovich and Charalambos D. Aliprantis, "An Invitation to Operator Theory", p.156
- ↑ T. Kato, "Perturbation theory for the nullity deficiency and other quantities of linear operators",
*J. d'Analyse Math*.**6**(1958), 273–322. - ↑ Berkani Mohammed: On a class of quasi-Fredholm operators INTEGRAL EQUATIONS AND OPERATOR THEORY Volume 34, Number 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"). - Template:Planetmath reference
- Weisstein, Eric W., "Fredholm's Theorem",
*MathWorld*. - {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- 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)