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

The index of a Fredholm operator is the integer

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

or in other words,

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

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

$S:Y\to X$

such that

${\mathrm{I}\mathrm{d}}_X-ST\text{and}{\mathrm{I}\mathrm{d}}_Y-TS$

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 $U\circ T$ is Fredholm from X to Z and

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

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. 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 $T\in B(X,Y)$ is inessential if and only if T+U is Fredholm for every Fredholm operator $U\in B(X,Y)$.

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

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

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

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

The left shift S* is Fredholm with index 1.

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

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

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

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

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

Semi-Fredholm operators

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

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

Unbounded operators

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

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

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)