Fredholm operator

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 ∘ T is Fredholm from X to Z and

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 ∈ B ( X , Y ) is strictly singular if and only if

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 ∈ B ( X , Y ) is inessential if and only if T+U is Fredholm for every Fredholm operator U ∈ 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

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,

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

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):

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.