Fredholm operator

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 $\ker T$ and finite-dimensional (algebraic) cokernel $\operatornameT = Y/\operatornameT$, and with closed range $\operatornameT$. The last condition is actually redundant.

The ''index'' of a Fredholm operator is the integer

:$\operatornameT := \dim \ker T - \operatorname\operatornameT$

or in other words,

:$\operatornameT := \dim \ker T - \operatorname\operatornameT.$

Properties

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

:$S: Y\to X$

such that

:$\mathrm_X - ST \quad\text\quad \mathrm_Y - TS$

are compact operators on $X$ and $Y$ respectively.

If a Fredholm operator is modified slightly, it stays Fredholm and its index remains the same. Formally: The set of Fredholm operators from $X$ to $Y$ is open in the Banach space $L(X,Y)$ of bounded linear operators, equipped with the operator norm, and the index is locally constant. More precisely, if $T_0$ is Fredholm from $X$ to $Y$, there exists $\varepsilon > 0$ such that every $T$ in $L(X,Y)$ with $, , T - T_0, , < \varepsilon$ is Fredholm, with the same index as that of $T_0$

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

:$\operatorname (U \circ T) = \operatorname(U) + \operatorname(T).$

When $T$ is Fredholm, the transpose (or adjoint) operator $T'$ is Fredholm from $Y'$ to $X'$, and $\text(T') = -\text(T)$. 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 such a compact perturbations of $T$. This follows from the fact that the index $\text(T+sK)$ is an integer defined for every $s$ in ,1/math>, and $\text(T+sK)$ is locally constant, hence $\text(T+sK) = \text(T+sK)$.

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. 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 $T\in B(X,Y)$ is inessential if and only if $T+U$ is Fredholm for every Fredholm operator $U\in B(X,Y)$.

Examples

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

:$S(e_n) = e_, \quad n \ge 0. \,$

This operator ''S'' is injective (actually, isometric) and has a closed range of codimension 1, hence ''S'' is Fredholm with $\operatorname(S)=-1$. The powers $S^k$, $k\geq0$, are Fredholm with index $-k$. The adjoint ''S*'' is the left shift,

:$S^*(e_0) = 0, \ \ S^*(e_n) = e_, \quad n \ge 1. \,$

The left shift ''S*'' is Fredholm with index 1.

If ''H'' is the classical Hardy space $H^2(\mathbf)$ on the unit circle T in the complex plane, then the shift operator with respect to the orthonormal basis of complex exponentials

:$e_n : \mathrm^ \in \mathbf \mapsto \mathrm^, \quad n \ge 0, \,$

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

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

Then ''T''_''φ'' is a Fredholm operator on $H^2(\mathbf)$, with index related to the winding number around 0 of the closed path t\in ,2\pimapsto \varphi(e^): 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 $\ker T$, $\operatornameT$ is finite-dimensional. For a semi-Fredholm operator, the index is defined by

:$\operatornameT=\begin +\infty,&\dim\ker T=\infty; \\ \dim\ker T-\dim\operatornameT,&\dim\ker T+\dim\operatornameT<\infty; \\ -\infty,&\dim\operatornameT=\infty. \end$

Unbounded operators

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

# The closed linear operator $T:\,X\to Y$ is called ''Fredholm'' if its domain $\mathfrak(T)$ is dense in $X$, its range is closed, and both kernel and cokernel of ''T'' are finite-dimensional.
#$T:\,X\to Y$ is called ''semi-Fredholm'' if its domain $\mathfrak(T)$ is dense in $X$, 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

References

* D.E. Edmunds and W.D. Evans (1987), ''Spectral theory and differential operators,'' Oxford University Press. .
* 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").
*
*
* 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)

{{DEFAULTSORT:Fredholm Operator
Fredholm theory
Linear operators