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Concept
Fredholm operator
Summary
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 and finite-dimensional (algebraic) cokernel , and with closed range . The last condition is actually redundant.
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." The formally correct statement follows. A bounded operator T : X → 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
such that
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 − 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 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 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.
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