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 ${\displaystyle V}$ is a ${\displaystyle n}$-dimensional vector space, with ${\displaystyle n}$ finite, and ${\displaystyle T:V\to V}$ is a linear transformation, then exactly one of the following holds:

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

A more elementary formulation, in terms of matrices, is as follows. Given an ${\displaystyle m\times n}$ matrix ${\displaystyle A}$ and a ${\displaystyle m\times 1}$ column vector ${\displaystyle \mathbf {b} }$, exactly one of the following must hold:

1. Either:${\displaystyle A\mathbf {x} =\mathbf {b} }$ has a solution ${\displaystyle \mathbf {x} }$
2. Or:${\displaystyle A^{T}\mathbf {y} =0}$ has a solution ${\displaystyle \mathbf {y} }$ with ${\displaystyle \mathbf {y} ^{T}\mathbf {b} \neq 0}$.

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

Integral equations

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

${\displaystyle \lambda \varphi (x)-\int _{a}^{b}K(x,y)\varphi (y)\,dy=0}$

and the inhomogeneous equation

${\displaystyle \lambda \varphi (x)-\int _{a}^{b}K(x,y)\varphi (y)\,dy=f(x).}$

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

A sufficient condition for this statement to be true is for ${\displaystyle K(x,y)}$ to be square integrable on the rectangle ${\displaystyle [a,b]\times [a,b]}$ (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) $${\displaystyle T=\lambda -K}$$ to mean $${\displaystyle T(x,y)=\lambda \;\delta (x-y)-K(x,y)}$$ with ${\displaystyle \delta (x-y)}$ the Dirac delta function, considered as a distribution, or generalized function, in two variables. Then by convolution, ${\displaystyle T}$ induces a linear operator acting on a Banach space ${\displaystyle V}$ of functions ${\displaystyle \varphi (x)}$$${\displaystyle V\to V}$$ given by $${\displaystyle \varphi \mapsto \psi }$$ with ${\displaystyle \psi }$ given by $${\displaystyle \psi (x)=\int _{a}^{b}T(x,y)\varphi (y)\,dy=\lambda \;\varphi (x)-\int _{a}^{b}K(x,y)\varphi (y)\,dy.}$$

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

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 ${\displaystyle L}$ would be the Laplacian plus some lower order terms. Combined with suitable boundary conditions and expressed on a suitable Banach space ${\displaystyle X}$ (which encodes both the boundary conditions and the desired regularity of the solution), ${\displaystyle L}$ becomes an unbounded operator from ${\displaystyle X}$ to itself, and one attempts to solve

${\displaystyle Lu=f,\qquad u\in \operatorname {dom} (L)\subseteq X,}$

where ${\displaystyle f\in X}$ 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

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

supplemented with the boundary condition

${\displaystyle (**)\qquad u=0\qquad {\text{on }}\partial \Omega ,}$

where ${\displaystyle \Omega \subseteq \mathbf {R} ^{n}}$ is a bounded open set with smooth boundary and ${\displaystyle h}$ is a fixed coefficient function (a potential, in the case of a Schrödinger operator). The function ${\displaystyle f\in X}$ is the variable data for which we wish to solve the equation. Here one would take ${\displaystyle X}$ to be the space ${\displaystyle L^{2}(\Omega )}$ of all square-integrable functions on ${\displaystyle \Omega }$, and ${\displaystyle \operatorname {dom} (L)}$ is then the Sobolev space ${\displaystyle W^{2,2}(\Omega )\cap W_{0}^{1,2}(\Omega )}$, which amounts to the set of all square-integrable functions on ${\displaystyle \Omega }$ whose weak first and second derivatives exist and are square-integrable, and which satisfy a zero boundary condition on ${\displaystyle \partial \Omega }$.

If ${\displaystyle X}$ has been selected correctly (as it has in this example), then for ${\displaystyle \mu _{0}\gg 0}$ the operator ${\displaystyle L+\mu _{0}}$ is positive, and then employing elliptic estimates, one can prove that ${\displaystyle L+\mu _{0}:\operatorname {dom} (L)\to X}$ is a bijection, and its inverse is a compact, everywhere-defined operator ${\displaystyle K}$ from ${\displaystyle X}$ to ${\displaystyle X}$, with image equal to ${\displaystyle \operatorname {dom} (L)}$. We fix one such ${\displaystyle \mu _{0}}$, 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 ${\displaystyle \lambda \in \mathbf {R} }$, either ${\displaystyle \lambda }$ is an eigenvalue of ${\displaystyle K}$, or the operator ${\displaystyle K-\lambda }$ is bijective from ${\displaystyle X}$ to itself.

Let us explore the two alternatives as they play out for the boundary-value problem. Suppose ${\displaystyle \lambda \neq 0}$. Then either

(A) ${\displaystyle \lambda }$ is an eigenvalue of ${\displaystyle K}$${\displaystyle \Leftrightarrow }$ there is a solution ${\displaystyle h\in \operatorname {dom} (L)}$ of ${\displaystyle (L+\mu _{0})h=\lambda ^{-1}h}$${\displaystyle \Leftrightarrow }$${\displaystyle -\mu _{0}+\lambda ^{-1}}$ is an eigenvalue of ${\displaystyle L}$.

(B) The operator ${\displaystyle K-\lambda :X\to X}$ is a bijection ${\displaystyle \Leftrightarrow }$${\displaystyle (K-\lambda )(L+\mu _{0})=\operatorname {Id} -\lambda (L+\mu _{0}):\operatorname {dom} (L)\to X}$ is a bijection ${\displaystyle \Leftrightarrow }$${\displaystyle L+\mu _{0}-\lambda ^{-1}:\operatorname {dom} (L)\to X}$ is a bijection.

Replacing ${\displaystyle -\mu _{0}+\lambda ^{-1}}$ by ${\displaystyle \lambda }$, and treating the case ${\displaystyle \lambda =-\mu _{0}}$ separately, this yields the following Fredholm alternative for an elliptic boundary-value problem:

• For each ${\displaystyle \lambda \in \mathbf {R} }$, either the homogeneous equation ${\displaystyle (L-\lambda )u=0}$ has a nontrivial solution, or the inhomogeneous equation ${\displaystyle (L-\lambda )u=f}$ possesses a unique solution ${\displaystyle u\in \operatorname {dom} (L)}$ for each given datum ${\displaystyle f\in X}$.

The latter function ${\displaystyle u}$ 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 ${\displaystyle \lambda }$ for which the solvability fails is a discrete subset of ${\displaystyle \mathbf {R} }$ (the eigenvalues of ${\displaystyle L}$). 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 .
• A. G. Ramm, "A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators", American Mathematical Monthly, 108 (2001) p. 855.
• Khvedelidze, B.V. (2001) [1994], "Fredholm theorems", Encyclopedia of Mathematics , EMS Press
• "Fredholm alternative", Encyclopedia of Mathematics , EMS Press, 2001 [1994]