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 ${\displaystyle H}$ be a Hilbert space with an orthonormal basis ${\displaystyle \{e_{n}\}}$ indexed by the non negative integers. The (right) 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).