Fredholm operator

Main article: Fredholm theory

In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator T : X → Y between two Banach spaces with finite-dimensional kernel ${\displaystyle \ker T}$ and finite-dimensional (algebraic) cokernel ${\displaystyle \operatorname {coker} T=Y/\operatorname {ran} T}$, and with closed range ${\displaystyle \operatorname {ran} T}$. The last condition is actually redundant.^[1]

The index of a Fredholm operator is the integer

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

or in other words,

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

Properties

Intuitively, Fredholm operators are those operators that are invertible "if finite-dimensional effects are ignored." The formally correct statement follows. A bounded operator ${\displaystyle T:X\to Y}$  between Banach spaces ${\displaystyle X}$  and ${\displaystyle Y}$  is Fredholm if and only if it is invertible modulo compact operators, i.e., if there exists a bounded linear operator

${\displaystyle S:Y\to X}$ 

such that

${\displaystyle \mathrm {Id} _{X}-ST\quad {\text{and}}\quad \mathrm {Id} _{Y}-TS}$ 

are compact operators on ${\displaystyle X}$  and ${\displaystyle Y}$  respectively.

If a Fredholm operator is modified slightly, it stays Fredholm and its index remains the same. Formally: The set of Fredholm operators from ${\displaystyle X}$  to ${\displaystyle Y}$  is open in the Banach space ${\displaystyle L(X,Y)}$  of bounded linear operators, equipped with the operator norm, and the index is locally constant. More precisely, if ${\displaystyle T_{0}}$  is Fredholm from ${\displaystyle X}$  to ${\displaystyle Y}$ , there exists ${\displaystyle \varepsilon >0}$  such that every ${\displaystyle T}$  in ${\displaystyle L(X,Y)}$  with ${\displaystyle ||T-T_{0}||<\varepsilon }$  is Fredholm, with the same index as that of ${\displaystyle T_{0}}$ 

When ${\displaystyle T}$  is Fredholm from ${\displaystyle X}$  to ${\displaystyle Y}$  and ${\displaystyle U}$  Fredholm from ${\displaystyle Y}$  to ${\displaystyle Z}$ , then the composition ${\displaystyle U\circ T}$  is Fredholm from ${\displaystyle X}$  to ${\displaystyle Z}$  and

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

When ${\displaystyle T}$  is Fredholm, the transpose (or adjoint) operator ${\displaystyle T'}$  is Fredholm from ${\displaystyle Y'}$  to ${\displaystyle X'}$ , and ${\displaystyle {\text{ind}}(T')=-{\text{ind}}(T)}$ . When ${\displaystyle X}$  and ${\displaystyle Y}$  are Hilbert spaces, the same conclusion holds for the Hermitian adjoint ${\displaystyle T^{*}}$ .

When ${\displaystyle T}$  is Fredholm and ${\displaystyle K}$  a compact operator, then ${\displaystyle T+K}$  is Fredholm. The index of ${\displaystyle T}$  remains unchanged under such a compact perturbations of ${\displaystyle T}$ . This follows from the fact that the index ${\displaystyle {\text{ind}}(T+sK)}$  is an integer defined for every ${\displaystyle s}$  in ${\displaystyle [0,1]}$ , and ${\displaystyle {\text{ind}}(T+sK)}$  is locally constant, hence ${\displaystyle {\text{ind}}(T+sK)={\text{ind}}(T+sK)}$ .

Invariance by perturbation is true for larger classes than the class of compact operators. For example, when ${\displaystyle U}$  is Fredholm and ${\displaystyle T}$  a strictly singular operator, then ${\displaystyle T+U}$  is Fredholm with the same index.^[2] 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 ${\displaystyle T+U}$  is Fredholm for every Fredholm operator ${\displaystyle U\in B(X,Y)}$ .

Examples

Let ${\displaystyle X}$  and ${\displaystyle Y}$  be Hilbert spaces. If ${\displaystyle T:X\to Y}$  is Fredholm, then we have a decomposition:$${\displaystyle X=\ker T\oplus (\ker T)^{\perp },\quad Y=\operatorname {ran} T\oplus (\operatorname {ran} T)^{\perp }}$$ ${\displaystyle T}$  is a bicontinuous bijection between ${\displaystyle (\ker T)^{\perp },\operatorname {ran} T}$ , and for it to be Fredholm, both ${\displaystyle \ker T,(\operatorname {ran} T)^{\perp }}$  must be finite-dimensional. By picking an appropriate orthonormal basis, ${\displaystyle T}$  has a matrix representation ${\displaystyle {\begin{bmatrix}0_{\operatorname {dim} \operatorname {coker} T\times \dim \ker T}&0\\0&\ddots \end{bmatrix}}}$ .

Let ${\displaystyle H}$  be a Hilbert space with an orthonormal basis ${\displaystyle \{e_{n}\}}$  indexed by the non negative integers. The unilateral shift operator S on H is defined by

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

This operator S is injective (actually, isometric) and has a closed range of codimension 1, hence S is Fredholm with ${\displaystyle \operatorname {ind} (S)=-1}$ . The powers ${\displaystyle S^{k}}$ , ${\displaystyle k\geq 0}$ , are Fredholm with index ${\displaystyle -k}$ . The adjoint S* is the left shift,

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

The left shift S* is Fredholm with index 1.

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

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

is the multiplication operator M_φ with the function ${\displaystyle \varphi =e_{1}}$ . More generally, let φ be a complex continuous function on T that does not vanish on ${\displaystyle \mathbf {T} }$ , and let T_φ denote the Toeplitz operator with symbol φ, equal to multiplication by φ followed by the orthogonal projection ${\displaystyle P:L^{2}(\mathbf {T} )\to H^{2}(\mathbf {T} )}$ :

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

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

Applications

Any elliptic operator on a closed manifold can be extended to a Fredholm operator. The use of Fredholm operators in partial differential equations is an abstract form of the parametrix method.

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

The Atiyah-Jänich theorem identifies the K-theory K(X) of a compact topological space X with the set of homotopy classes of continuous maps from X to the space of Fredholm operators H→H, where H is the separable Hilbert space and the set of these operators carries the operator norm.

Generalizations

Semi-Fredholm operators

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

${\displaystyle \operatorname {ind} T={\begin{cases}+\infty ,&\dim \ker T=\infty ;\\\dim \ker T-\dim \operatorname {coker} T,&\dim \ker T+\dim \operatorname {coker} T<\infty ;\\-\infty ,&\dim \operatorname {coker} T=\infty .\end{cases}}}$ 

Unbounded operators

One may also define unbounded Fredholm operators. Let X and Y be two Banach spaces.

1. The closed linear operator ${\displaystyle T:\,X\to Y}$  is called Fredholm if its ___domain ${\displaystyle {\mathfrak {D}}(T)}$  is dense in ${\displaystyle X}$ , its range is closed, and both kernel and cokernel of T are finite-dimensional.
2. ${\displaystyle T:\,X\to Y}$  is called semi-Fredholm if its ___domain ${\displaystyle {\mathfrak {D}}(T)}$  is dense in ${\displaystyle X}$ , its range is closed, and either kernel or cokernel of T (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).

Notes

 

The Wikibook Functional Analysis has a page on the topic of: Fredholm theory

1. Abramovich, Yuri A.; Aliprantis, Charalambos D. (2002). An Invitation to Operator Theory. Graduate Studies in Mathematics. Vol. 50. American Mathematical Society. p. 156. ISBN 978-0-8218-2146-6.
2. Kato, Tosio (1958). "Perturbation theory for the nullity deficiency and other quantities of linear operators". Journal d'Analyse Mathématique. 6: 273–322. doi:10.1007/BF02790238. S2CID 120480871.

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").
• Weisstein, Eric W. "Fredholm's Theorem". MathWorld.
• B.V. Khvedelidze (2001) [1994], "Fredholm theorems", Encyclopedia of Mathematics, EMS Press
• 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)