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 constant 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 T is Fredholm and S a strictly singular operator, then T + S is Fredholm with the same index. A bounded linear operator S from X to Y is strictly singular when its restriction to any infinite dimensional subspace X0 of X fails to be an into isomorphism, that is:
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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