Fredholm Alternative — Mathematics Knowledge System

Overview

The Fredholm alternative is a fundamental dichotomy for equations of the form (I - T)x = y where T is a compact operator. Either the homogeneous equation (I - T)x = 0 has only the trivial solution (and then the inhomogeneous equation has a unique solution for every y), or the homogeneous equation has a non-trivial finite-dimensional solution space (and then the inhomogeneous equation has solutions only for y orthogonal to the solutions of the adjoint homogeneous equation). This is the precise infinite-dimensional generalisation of the statement that a square matrix is either invertible or its null space is non-trivial.

Intuition

For a finite matrix A, either A is invertible (Ax = b has a unique solution for all b) or A is singular (Ax = 0 has non-trivial solutions and Ax = b has solutions only when b is orthogonal to the null space of A*). The Fredholm alternative says the same dichotomy holds for I - T when T is compact, even in infinite dimensions. The compactness of T is crucial: it makes I - T 'almost' a finite-dimensional perturbation of the identity.

Formal Definition

Definition

Let X be a Banach space and T: X -> X compact. For any lambda != 0, either: (i) I - lambda^{-1}T is bijective (the equation x - lambda^{-1}Tx = y has a unique solution for every y), or (ii) null(I - lambda^{-1}T) is finite-dimensional and non-trivial, and the equation x - lambda^{-1}Tx = y is solvable iff y is in (null(I - lambda^{-1}T*))^perp. In Hilbert space notation for T compact and mu = 1: either (I-T)x = y has a unique solution for all y, or ker(I-T) has finite dimension d >= 1, ker(I-T*) has the same dimension d, and (I-T)x = y is solvable iff y perp ker(I-T*).

\text{Either }\ker(I-T)=\{0\}\text{ and }(I-T)\text{ is surjective}

Case 1 (invertible)

\text{Or }\dim\ker(I-T)=\dim\ker(I-T^*)=d\ge 1

Case 2 (singular)

(I-T)x = y \text{ solvable} \iff y \perp \ker(I-T^*)

Solvability condition

Notation

Notation	Meaning
$\ker(I-T)$	Null space of I - T
$\operatorname{ran}(I-T)$	Range of I - T
$d$	Common dimension of ker(I-T) and ker(I-T*)
$\mathrm{ind}(T)$	Fredholm index: dim ker(T) - dim coker(T)

Theorems

Theorem 9.1: Theorem 9.1 (Fredholm Alternative)

Let H be a Hilbert space and T in K(H) compact. For any y in H, exactly one of the following holds: (i) x - Tx = y has a unique solution x for every y, or (ii) x - Tx = 0 has a non-trivial solution, the space ker(I-T) is finite-dimensional (say dimension d >= 1), ker(I-T*) also has dimension d, and x - Tx = y is solvable iff y perp ker(I-T*).

Theorem 9.2: Theorem 9.2 (Index zero)

For T compact, I - T is a Fredholm operator of index zero: dim ker(I-T) = dim coker(I-T) (where coker = X / ran(I-T)). In particular, I - T is injective iff it is surjective.

Theorem 9.3: Theorem 9.3 (Riesz–Schauder)

The non-zero spectrum of a compact operator T on an infinite-dimensional Banach space consists entirely of eigenvalues. Each non-zero eigenvalue has finite multiplicity and can accumulate only at 0.

Worked Examples

1
The equation is (I - T)u = f where T is the integral operator with kernel K. Since T is compact on L^2[0,1] (assuming K in L^2), the Fredholm alternative applies.
2
Case 1: If the homogeneous equation u - Tu = 0 has only u = 0, then for every f in L^2[0,1] there is a unique solution u.
3
Case 2: If u - Tu = 0 has non-trivial solutions phi_1,...,phi_d (a finite set), then u - Tu = f is solvable iff f is orthogonal to all solutions psi_1,...,psi_d of the adjoint equation v - T*v = 0:
$\int_0^1 f(x)\overline{\psi_j(x)}\,dx = 0 \quad j = 1,\ldots,d$

✓ Answer

The Fredholm integral equation (I-T)u = f has a unique solution for every f iff u - Tu = 0 has only the trivial solution; otherwise it is solvable iff f is orthogonal to all solutions of the adjoint homogeneous equation.

Practice Problems

Mediumproof writing

Prove that if T is compact and I - T is injective, then I - T is surjective.

Mediumfree response

State the Fredholm alternative and explain its significance for integral equations.

Quiz

Summary

Fredholm alternative: for compact T, (I-T)x = y either has a unique solution for all y, or is solvable only when y satisfies orthogonality conditions.
In Case 2: dim ker(I-T) = dim ker(I-T*) = d >= 1 (index zero property).
Solvability condition: (I-T)x = y is solvable iff y perp ker(I-T*).
Riesz–Schauder: non-zero spectrum of compact T consists only of eigenvalues of finite multiplicity.
Applications: Fredholm integral equations, Sturm–Liouville BVPs, elliptic PDE.

References

BookBrezis, H. — Functional Analysis, Sobolev Spaces and PDEs, Chapter 6
BookKreyszig, E. — Introductory Functional Analysis with Applications, Chapter 8
Websiteen.wikipedia.org