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 honor 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 ${\displaystyle \mathrm{coker}T=Y/\mathrm{ran}T}$, and with closed range ${\displaystyle \mathrm{ran}T}$. The last condition is actually redundant.^[1]

The index of a Fredholm operator is the integer

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

alternatively,

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

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

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

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 ${\displaystyle \epsilon >0}$ such that every ${\displaystyle T}$ in ${\displaystyle L(X,Y)}$ with ${\displaystyle ||T-T_0||<\epsilon }$ 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

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

When ${\displaystyle T}$ is Fredholm, the transpose (or adjoint) operator ${\displaystyle T^{\prime }}$ is Fredholm from ${\displaystyle Y^{\prime }}$ to ${\displaystyle X^{\prime }}$, and ${\displaystyle \text{ind}(T^{\prime })=-\text{ind}(T)}$. When ${\displaystyle X}$ and ${\displaystyle Y}$ are Hilbert spaces, the same conclusion holds for the Hermitian adjoint ${\displaystyle T^{\ast }}$.

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 perturbation 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 ${\displaystyle \text{ind}(T)=\text{ind}(T+sK)}$.

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:$${\displaystyle X=\ker T\oplus (\ker T{)}^{\perp },Y=\mathrm{ran}T\oplus (\mathrm{ran}T{)}^{\perp }}$$${\displaystyle T}$ is a bicontinuous bijection between ${\displaystyle (\ker T{)}^{\perp },\mathrm{ran}T}$, and for it to be Fredholm, both ${\displaystyle \ker T,(\mathrm{ran}T{)}^{\perp }}$ must be finite-dimensional. By picking an appropriate orthonormal basis, ${\displaystyle T}$ has a matrix representation ${\displaystyle \left[\begin{array}{cc}{0}_{\dim \mathrm{coker}T\times \dim \ker T}& 0\\ 0& \ddots \end{array}\right]}$.

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

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

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

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

The left shift S* is Fredholm with index 1.

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

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

is the multiplication operator M_φ with the function ${\displaystyle \phi =e_1}$. More generally, let φ be a complex continuous function on T that does not vanish on ${\displaystyle 𝐓}$, and let T_φ denote the Toeplitz operator with symbol φ, equal to multiplication by φ followed by the orthogonal projection ${\displaystyle P:L^2(𝐓)\to H^2(𝐓)}$:

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

Then T_φ is a Fredholm operator on ${\displaystyle H^2(𝐓)}$, with index related to the winding number around 0 of the closed path ${\displaystyle t\in [0,2\pi ]\mapsto \phi (e^{it})}$: 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 ${\displaystyle \ker T}$, ${\displaystyle \mathrm{coker}T}$ is finite-dimensional. For a semi-Fredholm operator, the index is defined by

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

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 ${\displaystyle 𝔇(T)}$ 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 ${\displaystyle 𝔇(T)}$ 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

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)