Fredholm alternative

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

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

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

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

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

Integral equations

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

λ φ ( x ) − ∫ a b K ( x , y ) φ ( y ) d y = 0 {\displaystyle \lambda \varphi (x)-\int _{a}^{b}K(x,y)\varphi (y)\,dy=0} {\displaystyle \lambda \varphi (x)-\int _{a}^{b}K(x,y)\varphi (y)\,dy=0}

and the inhomogeneous equation

λ φ ( x ) − ∫ a b K ( x , y ) φ ( y ) d y = f ( x ) . {\displaystyle \lambda \varphi (x)-\int _{a}^{b}K(x,y)\varphi (y)\,dy=f(x).} {\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 λ ∈ C , {\displaystyle \lambda \in \mathbb {C} ,} {\displaystyle \lambda \in \mathbb {C} ,} either the first equation has a non-trivial solution, or the second equation has a solution for all f ( x ) {\displaystyle f(x)} {\displaystyle f(x)}.

A sufficient condition for this statement to be true is for K ( x , y ) {\displaystyle K(x,y)} {\displaystyle K(x,y)} to be square integrable on the rectangle [ a , b ] × [ a , b ] {\displaystyle [a,b]\times [a,b]} {\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) T = λ − K {\displaystyle T=\lambda -K} {\displaystyle T=\lambda -K} to mean T ( x , y ) = λ δ ( x − y ) − K ( x , y ) {\displaystyle T(x,y)=\lambda \;\delta (x-y)-K(x,y)} {\displaystyle T(x,y)=\lambda \;\delta (x-y)-K(x,y)} with δ ( x − y ) {\displaystyle \delta (x-y)} {\displaystyle \delta (x-y)} the Dirac delta function, considered as a distribution, or generalized function, in two variables. Then by convolution, T {\displaystyle T} {\displaystyle T} induces a linear operator acting on a Banach space V {\displaystyle V} {\displaystyle V} of functions φ ( x ) {\displaystyle \varphi (x)} {\displaystyle \varphi (x)} V → V {\displaystyle V\to V} {\displaystyle V\to V} given by φ ↦ ψ {\displaystyle \varphi \mapsto \psi } {\displaystyle \varphi \mapsto \psi } with ψ {\displaystyle \psi } {\displaystyle \psi } given by ψ ( x ) = ∫ a b T ( x , y ) φ ( y ) d y = λ φ ( x ) − ∫ a b K ( x , y ) φ ( y ) d y . {\displaystyle \psi (x)=\int _{a}^{b}T(x,y)\varphi (y)\,dy=\lambda \;\varphi (x)-\int _{a}^{b}K(x,y)\varphi (y)\,dy.} {\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 K {\displaystyle K} {\displaystyle K} given by convolution with an L 2 {\displaystyle L^{2}} {\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 K {\displaystyle K} {\displaystyle K} is any compact operator. The Fredholm alternative may be restated in the following form: a nonzero λ {\displaystyle \lambda } {\displaystyle \lambda } either is an eigenvalue of K , {\displaystyle K,} {\displaystyle K,} or lies in the domain of the resolvent R ( λ ; K ) = ( K − λ Id ) − 1 . {\displaystyle R(\lambda ;K)=(K-\lambda \operatorname {Id} )^{-1}.} {\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 L {\displaystyle L} {\displaystyle L} would be the Laplacian plus some lower order terms. Combined with suitable boundary conditions and expressed on a suitable Banach space X {\displaystyle X} {\displaystyle X} (which encodes both the boundary conditions and the desired regularity of the solution), L {\displaystyle L} {\displaystyle L} becomes an unbounded operator from X {\displaystyle X} {\displaystyle X} to itself, and one attempts to solve

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

where f ∈ X {\displaystyle f\in X} {\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

( ∗ ) L u := − Δ u + h ( x ) u = f in  Ω , {\displaystyle (*)\qquad Lu:=-\Delta u+h(x)u=f\qquad {\text{in }}\Omega ,} {\displaystyle (*)\qquad Lu:=-\Delta u+h(x)u=f\qquad {\text{in }}\Omega ,}

supplemented with the boundary condition

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

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

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

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

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

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

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

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

The latter function u {\displaystyle u} {\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 λ {\displaystyle \lambda } {\displaystyle \lambda } for which the solvability fails is a discrete subset of R {\displaystyle \mathbf {R} } {\displaystyle \mathbf {R} } (the eigenvalues of L {\displaystyle L} {\displaystyle L}). The eigenvalues’ associated eigenfunctions can be thought of as "resonances" that block the solvability of the equation.

See also

References