Fredholm alternative

In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators. Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue.

Linear algebra

If V is an n-dimensional vector space and [math]\displaystyle{ T:V\to V }[/math] is a linear transformation, then exactly one of the following holds:

1. For each vector v in V there is a vector u in V so that [math]\displaystyle{ T(u) = v }[/math]. In other words: T is surjective (and so also bijective, since V is finite-dimensional).
2. [math]\displaystyle{ \dim(\ker(T)) \gt 0. }[/math]

A more elementary formulation, in terms of matrices, is as follows. Given an m×n matrix A and a m×1 column vector b, exactly one of the following must hold:

1. Either: A x = b has a solution x
2. Or: A^T y = 0 has a solution y with y^Tb ≠ 0.

In other words, A x = b has a solution [math]\displaystyle{ (\mathbf{b} \in \operatorname{Im}(A)) }[/math] if and only if for any y such that A^T y = 0, it follows that y^Tb = 0 [math]\displaystyle{ (i.e., \mathbf{b} \in \ker(A^T)^{\bot}) }[/math].

Integral equations

Let [math]\displaystyle{ K(x,y) }[/math] be an integral kernel, and consider the homogeneous equation, the Fredholm integral equation,

[math]\displaystyle{ \lambda \varphi(x)- \int_a^b K(x,y) \varphi(y) \,dy = 0 }[/math]

and the inhomogeneous equation

[math]\displaystyle{ \lambda \varphi(x) - \int_a^b K(x,y) \varphi(y) \,dy = f(x). }[/math]

The Fredholm alternative is the statement that, for every non-zero fixed complex number [math]\displaystyle{ \lambda \in \mathbb{C}, }[/math] either the first equation has a non-trivial solution, or the second equation has a solution for all [math]\displaystyle{ f(x) }[/math].

A sufficient condition for this statement to be true is for [math]\displaystyle{ K(x,y) }[/math] to be square integrable on the rectangle [math]\displaystyle{ [a,b]\times[a,b] }[/math] (where a and/or b may be minus or plus infinity). The integral operator defined by such a K is called a Hilbert–Schmidt integral operator.

Functional analysis

Results about Fredholm operators generalize these results to complete normed vector spaces of infinite dimensions; that is, Banach spaces.

The integral equation can be reformulated in terms of operator notation as follows. Write (somewhat informally) [math]\displaystyle{ T = \lambda - K }[/math] to mean [math]\displaystyle{ T(x,y) = \lambda\; \delta(x-y) - K(x,y) }[/math] with [math]\displaystyle{ \delta(x-y) }[/math] the Dirac delta function, considered as a distribution, or generalized function, in two variables. Then by convolution, [math]\displaystyle{ T }[/math] induces a linear operator acting on a Banach space [math]\displaystyle{ V }[/math] of functions [math]\displaystyle{ \varphi(x) }[/math] [math]\displaystyle{ V \to V }[/math] given by [math]\displaystyle{ \varphi \mapsto \psi }[/math] with [math]\displaystyle{ \psi }[/math] given by [math]\displaystyle{ \psi(x)=\int_a^b T(x,y) \varphi(y) \,dy = \lambda\;\varphi(x) - \int_a^b K(x,y) \varphi(y) \,dy. }[/math]

In this language, the Fredholm alternative for integral equations is seen to be analogous to the Fredholm alternative for finite-dimensional linear algebra.

The operator [math]\displaystyle{ K }[/math] given by convolution with an [math]\displaystyle{ L^2 }[/math] kernel, as above, is known as a Hilbert–Schmidt integral operator. Such operators are always compact. More generally, the Fredholm alternative is valid when [math]\displaystyle{ K }[/math] is any compact operator. The Fredholm alternative may be restated in the following form: a nonzero [math]\displaystyle{ \lambda }[/math] either is an eigenvalue of [math]\displaystyle{ K, }[/math] or lies in the domain of the resolvent [math]\displaystyle{ R(\lambda; K) = (K-\lambda \operatorname{Id})^{-1}. }[/math]

Elliptic partial differential equations

The Fredholm alternative can be applied to solving linear elliptic boundary value problems. The basic result is: if the equation and the appropriate Banach spaces have been set up correctly, then either

(1) The homogeneous equation has a nontrivial solution, or

(2) The inhomogeneous equation can be solved uniquely for each choice of data.

The argument goes as follows. A typical simple-to-understand elliptic operator L would be the Laplacian plus some lower order terms. Combined with suitable boundary conditions and expressed on a suitable Banach space X (which encodes both the boundary conditions and the desired regularity of the solution), L becomes an unbounded operator from X to itself, and one attempts to solve

[math]\displaystyle{ L u = f,\qquad u\in \operatorname{dom}(L) \subseteq X, }[/math]

where f ∈ X is some function serving as data for which we want a solution. The Fredholm alternative, together with the theory of elliptic equations, will enable us to organize the solutions of this equation.

A concrete example would be an elliptic boundary-value problem like

[math]\displaystyle{ (*)\qquad Lu := -\Delta u + h(x) u = f\qquad \text{in }\Omega, }[/math]

supplemented with the boundary condition

[math]\displaystyle{ (**) \qquad u = 0 \qquad \text{on } \partial\Omega, }[/math]

where Ω ⊆ Rⁿ is a bounded open set with smooth boundary and h(x) is a fixed coefficient function (a potential, in the case of a Schrödinger operator). The function f ∈ X is the variable data for which we wish to solve the equation. Here one would take X to be the space L²(Ω) of all square-integrable functions on Ω, and dom(L) is then the Sobolev space W ^2,2(Ω) ∩ W1,20(Ω), which amounts to the set of all square-integrable functions on Ω whose weak first and second derivatives exist and are square-integrable, and which satisfy a zero boundary condition on ∂Ω.

If X has been selected correctly (as it has in this example), then for μ₀ >> 0 the operator L + μ₀ is positive, and then employing elliptic estimates, one can prove that L + μ₀ : dom(L) → X is a bijection, and its inverse is a compact, everywhere-defined operator K from X to X, with image equal to dom(L). We fix one such μ₀, but its value is not important as it is only a tool.

We may then transform the Fredholm alternative, stated above for compact operators, into a statement about the solvability of the boundary-value problem (*)–(**). The Fredholm alternative, as stated above, asserts:

• For each λ ∈ R, either λ is an eigenvalue of K, or the operator K − λ is bijective from X to itself.

Let us explore the two alternatives as they play out for the boundary-value problem. Suppose λ ≠ 0. Then either

(A) λ is an eigenvalue of K ⇔ there is a solution h ∈ dom(L) of (L + μ₀) h = λ⁻¹h ⇔ –μ₀+λ⁻¹ is an eigenvalue of L.

(B) The operator K − λ : X → X is a bijection ⇔ (K − λ) (L + μ₀) = Id − λ (L + μ₀) : dom(L) → X is a bijection ⇔ L + μ₀ − λ⁻¹ : dom(L) → X is a bijection.

Replacing -μ₀+λ⁻¹ by λ, and treating the case λ = −μ₀ separately, this yields the following Fredholm alternative for an elliptic boundary-value problem:

• For each λ ∈ R, either the homogeneous equation (L − λ) u = 0 has a nontrivial solution, or the inhomogeneous equation (L − λ) u = f possesses a unique solution u ∈ dom(L) for each given datum f ∈ X.

The latter function u solves the boundary-value problem (*)–(**) introduced above. This is the dichotomy that was claimed in (1)–(2) above. By the spectral theorem for compact operators, one also obtains that the set of λ for which the solvability fails is a discrete subset of R (the eigenvalues of L). The eigenvalues’ associated eigenfunctions can be thought of as "resonances" that block the solvability of the equation.

See also

• Spectral theory of compact operators
• Farkas' lemma

References

• Fredholm, E. I. (1903). "Sur une classe d'equations fonctionnelles". Acta Math. 27: 365–390. doi:10.1007/bf02421317. https://zenodo.org/record/1925972. 
• A. G. Ramm, "A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators", American Mathematical Monthly, 108 (2001) p. 855.
• Hazewinkel, Michiel, ed. (2001), "Fredholm theorems", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=f/f041470 
• Hazewinkel, Michiel, ed. (2001), "Fredholm alternative", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page