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
[1] The index of a Fredholm operator is the integer or in other words, Intuitively, Fredholm operators are those operators that are invertible "if finite-dimensional effects are ignored."
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.
When X and Y are Hilbert spaces, the same conclusion holds for the Hermitian adjoint T∗.
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.
[2] The class of inessential operators, which properly contains the class of strictly singular operators, is the "perturbation class" for Fredholm operators.
be a Hilbert space with an orthonormal basis
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
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 Mφ with the function
More generally, let φ be a complex continuous function on T that does not vanish on
, and let Tφ denote the Toeplitz operator with symbol φ, equal to multiplication by φ followed by the orthogonal projection
, with index related to the winding number around 0 of the closed path
: the index of Tφ, as defined in this article, is the opposite of this winding number.
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.
A bounded linear operator T is called semi-Fredholm if its range is closed and at least one of
For a semi-Fredholm operator, the index is defined by One may also define unbounded Fredholm operators.
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).