Fredholm operator

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(X, Y) 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(X, Y) with ||T − T₀|| < 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 + 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 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 M_φ 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 T_φ 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 T_φ 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 T_φ, 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 H→H, 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

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)