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

The index of a Fredholm operator is the integer


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

or in other words,


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

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 &rarr; 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


 * $$ \mathrm{Id}_X - ST \quad\text{and}\quad \mathrm{Id}_Y - TS $$

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 &minus; 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


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

When T is Fredholm, the transpose (or adjoint) operator T&thinsp;&prime; is Fredholm from Y&thinsp;&prime; to X&thinsp;&prime;, and ind(T&thinsp;&prime;) = &minus;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&thinsp;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


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

This operator S is injective (actually, isometric) and has a closed range of codimension 1, hence S is Fredholm with $$\operatorname{ind}(S)=-1$$. The powers $$S^k$$, $$k\geq0$$, 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 $$H^2(\mathbf{T})$$ 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} n t}, \quad n \ge 0, \, $$

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


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

Then Tφ is a Fredholm operator on $$H^2(\mathbf{T})$$, with index related to the winding number around 0 of the closed path $$t\in[0,2\pi]\mapsto \varphi(e^{it})$$: the index of  Tφ, 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&rarr;H, where H is the separable Hilbert space and the set of these operators carries the operator norm.

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



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

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.
 * $$T:\,X\to Y$$ 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).