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 [math]\displaystyle{ \ker T }[/math] and finite-dimensional (algebraic) cokernel [math]\displaystyle{ \operatorname{coker}T = Y/\operatorname{ran}T }[/math], and with closed range [math]\displaystyle{ \operatorname{ran}T }[/math]. The last condition is actually redundant.

The index of a Fredholm operator is the integer

[math]\displaystyle{ \operatorname{ind}T := \dim \ker T - \operatorname{codim}\operatorname{ran}T }[/math]

or in other words,

[math]\displaystyle{ \operatorname{ind}T := \dim \ker T - \operatorname{dim}\operatorname{coker}T. }[/math]

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

[math]\displaystyle{ S: Y\to X }[/math]

such that

[math]\displaystyle{ \mathrm{Id}_X - ST \quad\text{and}\quad \mathrm{Id}_Y - TS }[/math]

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 [math]\displaystyle{ U \circ T }[/math] is Fredholm from X to Z and

[math]\displaystyle{ \operatorname{ind} (U \circ T) = \operatorname{ind}(U) + \operatorname{ind}(T). }[/math]

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. 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 [math]\displaystyle{ T\in B(X,Y) }[/math] is inessential if and only if T+U is Fredholm for every Fredholm operator [math]\displaystyle{ U\in B(X,Y) }[/math].

Examples

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

[math]\displaystyle{ S(e_n) = e_{n+1}, \quad n \ge 0. \, }[/math]

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

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

The left shift S* is Fredholm with index 1.

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

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

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

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

Then T_φ is a Fredholm operator on [math]\displaystyle{ H^2(\mathbf{T}) }[/math], with index related to the winding number around 0 of the closed path [math]\displaystyle{ t\in[0,2\pi]\mapsto \varphi(e^{it}) }[/math]: 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

Semi-Fredholm operators

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

[math]\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} }[/math]

Unbounded operators

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

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

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". http://mathworld.wolfram.com/FredholmsTheorem.html. 
• Hazewinkel, Michiel, ed. (2001), "Fredholm theorems", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=f/f041470 
• 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)