Fredholm operator

Main article: Fredholm theory

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 T : X → Y between two Banach spaces with finite-dimensional kernel $\ker T$ and (algebraic) cokernel $\mathrm {coker} \,T=Y/\mathrm {ran} \,T$, and with closed range $\mathrm {ran} \,T$. The last condition is actually redundant.

Equivalently, an operator T : X → Y is Fredholm if it is invertible modulo compact operators, i.e., if there exists a bounded linear operator

$S:Y\to X$

such that

$\mathrm {Id} _{X}-ST\quad {\text{and}}\quad \mathrm {Id} _{Y}-TS$

are compact operators on X and Y respectively.

The index of a Fredholm operator is

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

or in other words,

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

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 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 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. A bounded linear operator T from X into Y is strictly singular when it fails to be bounded below on any infinite-dimensional subspace. In symbols, an operator $T\in B(X,Y)$ is strictly singular if and only if

$\inf\{\|Tx\|:x\in X_{0},\,\|x\|=1\}=0.\,$

for each infinite-dimensional subspace $X_{0}$ of $X$. 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

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

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 $S^{k}$, $k\geq 0$, are Fredholm with index $-k$. The adjoint S* is the left shift,

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

The left shift S* is Fredholm with index 1.

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

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

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

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

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

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

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").
• "Fredholm operator". PlanetMath.
• Weisstein, Eric W. "Fredholm's Theorem". MathWorld.
• B.V. Khvedelidze (2001) [1994], "Fredholm theorems", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
• 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)