Fredholm operator

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, with finite-dimensional kernel and cokernel, and with closed range. (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

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 $U\circ T$ 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 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.^[2] 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 ${\displaystyle T\in B(X,Y)}$ is strictly singular if and only if

${\displaystyle \inf\{\|Tx\|:x\in X_{0},\,\|x\|=1\}=0.\,}$

for each infinite-dimensional subspace $X_{0}$ of $X$. 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 ${\displaystyle T\in B(X,Y)}$ is inessential if and only if T+U is Fredholm for every Fredholm operator ${\displaystyle U\in B(X,Y)}$.

Examples

Let H be a Hilbert space with an orthonormal basis {e_n} indexed by the non negative integers. The (right) shift operator S on H is defined by

$S(e_{n})=e_{{n+1}},\quad n\geq 0.\,$

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 S^k, k ≥ 0, are Fredholm with index −k. The adjoint S^∗ is the left shift,

$S^{*}(e_{0})=0,\ \ S^{*}(e_{n})=e_{{n-1}},\quad n\geq 1.\,$

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

$e_{n}:{\mathrm {e}}^{{{\mathrm {i}}t}}\in {\mathbf {T}}\rightarrow {\mathrm {e}}^{{{\mathrm {i}}nt}},\quad n\geq 0,\,$

is the multiplication operator M_φ with the function φ = e₁. More generally, let φ be a complex continuous function on T that does not vanish on T, and let T_φ 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 T_φ 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 T_φ, as defined in this article, is the opposite of this winding number.

Applications

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

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

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, 34, 2 (1999), 244-249

References

• D.E. Edmunds and W.D. Evans (1987), Spectral theory and differential operators, Oxford University Press. ISBN 0-19-853542-2.
• A. G. Ramm, "A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators", American Mathematical Monthly, 108 (2001) p. 855 (NB: In this paper the word "Fredholm operator" refers to "Fredholm operator of index 0").
• "Fredholm operator". PlanetMath.
• Weisstein, Eric W. "Fredholm's Theorem". MathWorld.

• B.V. Khvedelidze (2001), "Fredholm theorems", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Bruce K. Driver, "Compact and Fredholm Operators and the Spectral Theorem", Analysis Tools with Applications, Chapter 35, pp. 579–600.
• Robert C. McOwen, "Fredholm theory of partial differential equations on complete Riemannian manifolds", Pacific J. Math. 87, no. 1 (1980), 169–185.
• Tomasz Mrowka, A Brief Introduction to Linear Analysis: Fredholm Operators, Geometry of Manifolds, Fall 2004 (Massachusetts Institute of Technology: MIT OpenCouseWare)