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
are compact operators on X and Y respectively.
The index of a Fredholm operator is
or in other words,
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 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 is strictly singular if and only if
for each infinite-dimensional subspace of . 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 is inessential if and only if T+U is Fredholm for every Fredholm operator .
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 Sk, 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 Mφ with the function φ = e1. More generally, let φ be a complex continuous function on T that does not vanish on T, and let Tφ 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 Tφ 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 Tφ, as defined in this article, is the opposite of this winding number.
The Atiyah-Singer index theorem gives a topological characterization of the index of certain operators on manifolds.
For each integer , define to be the restriction of to viewed as a map from into ( in particular ). If for some integer the space is closed and is a Fredholm operator,then is called a B-Fredholm operator. The index of a B-Fredholm operator is defined as the index of the Fredholm operator . It is shown that the index is independent of the integer . B-Fredholm operators were introduced by M. Berkani in 1999 as a generalization of Fredholm operators.
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