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.) 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 T is Fredholm and S a strictly singular operator, then T + S is Fredholm with the same index. A bounded linear operator S from X to Y is strictly singular when its restriction to any infinite dimensional subspace X₀ of X fails to be an into isomorphism, that is:

$\inf \{ \|S x\| : x \in X_0, \, \|x\| = 1 \} = 0. \,$

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

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

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,

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

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

If H is the classical Hardy space H²(T) on the unit circle T in the complex plane, then the shift operator with respect to the orthonormal basis of complex exponentials

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

is the multiplication operator M_φ with the function φ = e₁. 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 L²(T) onto H²(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 H²(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 $n$, 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 at PlanetMath.org.
• Weisstein, Eric W., "Fredholm's Theorem", MathWorld.
• 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)