Fredholm operator

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In mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations. It is named in honour of Erik Ivar Fredholm.

A Fredholm operator is a bounded linear operator between two Banach spaces whose kernel and cokernel are finite-dimensional and whose range is closed. (The last condition is actually redundant.[1]) Equivalently, an operator T : X → Y is Fredholm 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.

The index of a Fredholm operator is

 \mathrm{ind}\,T := \dim \ker T - \mathrm{codim}\,\mathrm{ran}\,T

or in other words,

 \mathrm{ind}\,T := \dim \ker T - \mathrm{dim}\,\mathrm{coker}\,T;

see dimension, kernel, codimension, range, and cokernel.

Properties[edit]

The set of Fredholm operators from X to Y is open in the Banach space L(XY) 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(XY) with ||TT0|| < 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

\mathrm{ind} (U \circ T) = \mathrm{ind}(U) + \mathrm{ind}(T).

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 + sK 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.[2] 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:

\inf \{ \|S x\| : x \in X_0, \, \|x\| = 1 \} = 0. \,

Examples[edit]

Let H be a Hilbert space with an orthonormal basis {en} 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 ind(S) = −1. The powers Sk, k ≥ 0, 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 H2(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} \rightarrow
\mathrm{e}^{\mathrm{i} n t}, \quad n \ge 0, \,

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):

 T_\varphi : f \in H^2(\mathrm{T}) \rightarrow P(f \varphi) \in H^2(\mathrm{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.

Applications[edit]

The Atiyah-Singer index theorem gives a topological characterization of the index of certain operators on manifolds.

An 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.

B-Fredholm operators[edit]

For each integer n, define  T_{n} to be the restriction of T to  R(T^{n}) viewed as a map from  R(T^{n}) into  R(T^{n}) ( in particular  T_{0} = T). If for some integer n the space  R(T^{n}) is closed and  T_{n} is a Fredholm operator,then T is called a B-Fredholm operator. The index of a B-Fredholm operator T is defined as the index of the Fredholm operator  T_n . It is shown that the index is independent of the integer  n. B-Fredholm operators were introduced by M. Berkani in 1999 as a generalization of Fredholm operators. [3]

Notes[edit]

  1. ^ Yuri A. Abramovich and Charalambos D. Aliprantis, "An Invitation to Operator Theory", p.156
  2. ^ T. Kato, "Perturbation theory for the nullity deficiency and other quantities of linear operators", J. d'Analyse Math. 6 (1958), 273–322.
  3. ^ Berkani Mohammed: On a class of quasi-Fredholm operators INTEGRAL EQUATIONS AND OPERATOR THEORY Volume 34, Number 2 (1999), 244-249 [1]

References[edit]