Inmathematics,Fredholm operators are certainoperators that arise in theFredholm theory ofintegral equations. They are named in honour ofErik Ivar Fredholm. By definition, a Fredholm operator is abounded linear operatorT : X → Y between twoBanach spaces with finite-dimensionalkernel${\displaystyle \ker T}$ and finite-dimensional (algebraic)cokernel${\displaystyle \mathrm{coker}T=Y/\mathrm{ran}T}$, and with closedrange${\displaystyle \mathrm{ran}T}$. The last condition is actually redundant.^[1]

Theindex of a Fredholm operator is the integer

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

or in other words,

${\displaystyle \mathrm{ind}T:=\dim \ker T-\dim \mathrm{coker}T.}$

Intuitively, Fredholm operators are those operators that are invertible "if finite-dimensional effects are ignored." The formally correct statement follows. A bounded operatorT : X → Y between Banach spacesX andY is Fredholm if and only if it is invertiblemodulocompact operators, i.e., if there exists a bounded linear operator

${\displaystyle S:Y\to X}$ 

such that

${\displaystyle {\mathrm{I}\mathrm{d}}_X-ST\text{and}{\mathrm{I}\mathrm{d}}_Y-TS}$ 

are compact operators onX andY respectively.

If a Fredholm operator is modified slightly, it stays Fredholm and its index remains the same. Formally: The set of Fredholm operators fromX toY is open in the Banach space L(X, Y) of bounded linear operators, equipped with theoperator norm, and the index is locally constant. More precisely, ifT₀ is Fredholm fromX toY, there existsε > 0 such that everyT in L(X, Y) with||T −T₀|| <ε is Fredholm, with the same index as that of T₀.

WhenT is Fredholm fromX toY andU Fredholm fromY toZ, then the composition${\displaystyle U\circ T}$  is Fredholm fromX toZ and

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

WhenT is Fredholm, thetranspose (or adjoint) operatorT ′ is Fredholm fromY ′ toX ′, andind(T ′) = −ind(T). WhenX andY areHilbert spaces, the same conclusion holds for theHermitian adjoint T^∗.

WhenT is Fredholm andK a compact operator, thenT + K is Fredholm. The index ofT remains unchanged under such a compact perturbations ofT. This follows from the fact that the indexi(s) ofT +s K is an integer defined for everys in [0, 1], andi(s) is locally constant, hencei(1) = i(0).

Invariance by perturbation is true for larger classes than the class of compact operators. For example, whenU is Fredholm andT astrictly singular operator, thenT + U is Fredholm with the same index.^[2] The class ofinessential 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 ifT+U is Fredholm for every Fredholm operator${\displaystyle U\in B(X,Y)}$ .

Let${\displaystyle H}$  be aHilbert space with an orthonormal basis${\displaystyle \{e_n\}}$  indexed by the non negative integers. The (right)shift operatorS onH is defined by

${\displaystyle S(e_n)=e_{n+1},n\ge 0.}$ 

This operatorS is injective (actually, isometric) and has a closed range of codimension 1, henceS is Fredholm with${\displaystyle \mathrm{ind}(S)=-1}$ . The powers${\displaystyle S^k}$ ,${\displaystyle k\ge 0}$ , are Fredholm with index${\displaystyle -k}$ . The adjointS* is the left shift,

${\displaystyle S^{\ast }(e_0)=0,S^{\ast }(e_n)=e_{n-1},n\ge 1.}$ 

The left shiftS* is Fredholm with index 1.

IfH is the classicalHardy space${\displaystyle H^2(\mathbf{T})}$  on the unit circleT in the complex plane, then the shift operator with respect to the orthonormal basis of complex exponentials

${\displaystyle e_n:{\mathrm{e}}^{\mathrm{i}t}\in \mathbf{T}\mapsto {\mathrm{e}}^{\mathrm{i}nt},n\ge 0,}$ 

is the multiplication operatorM_φ with the function${\displaystyle \phi =e_1}$ . More generally, letφ be a complex continuous function onT that does not vanish on${\displaystyle \mathbf{T}}$ , and letT_φ denote theToeplitz operator with symbolφ, equal to multiplication byφ followed by the orthogonal projection${\displaystyle P:L^2(\mathbf{T})\to H^2(\mathbf{T})}$ :

${\displaystyle T_{\phi }:f\in H^2(\mathrm{T})\mapsto P(f\phi )\in H^2(\mathrm{T}).}$ 

ThenT_φ is a Fredholm operator on${\displaystyle H^2(\mathbf{T})}$ , with index related to thewinding number around 0 of the closed path${\displaystyle t\in [0,2\pi ]\mapsto \phi (e^{it})}$ : the index ofT_φ, as defined in this article, is the opposite of this winding number.

Anyelliptic operator on a closed manifold can be extended to a Fredholm operator. The use of Fredholm operators inpartial differential equations is an abstract form of theparametrix method.

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

TheAtiyah-Jänich theorem identifies theK-theoryK(X) of a compact topological spaceX with the set ofhomotopy classes of continuous maps fromX to the space of Fredholm operatorsH→H, whereH is the separable Hilbert space and the set of these operators carries the operator norm.

A bounded linear operatorT is calledsemi-Fredholm if its range is closed and at least one of${\displaystyle \ker T}$ ,${\displaystyle \mathrm{coker}T}$  is finite-dimensional. For a semi-Fredholm operator, the index is defined by

 

One may also define unbounded Fredholm operators. LetX andY be two Banach spaces.

1. Theclosed linear operator${\displaystyle T:X\to Y}$  is calledFredholm if its domain${\displaystyle \mathfrak{D}(T)}$  is dense in${\displaystyle X}$ , its range is closed, and both kernel and cokernel ofT are finite-dimensional.
2. ${\displaystyle T:X\to Y}$  is calledsemi-Fredholm if its domain${\displaystyle \mathfrak{D}(T)}$  is dense in${\displaystyle X}$ , its range is closed, and either kernel or cokernel ofT (or both) is finite-dimensional.

As it was noted above, the range of a closed operator is closed as long as the cokernel is finite-dimensional (Edmunds and Evans, Theorem I.3.2).

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