Integral equations Systems of integral equations Theorem

Matrix Wiener–Hopf factorization

Builds on Wiener–Hopf factorization System of linear integral equations Fourier transform Invertible matrix Winding number Riemann–Hilbert problem Wiener–Hopf equation

Statement

Let $n \ge 1$ and let $K$ be an $n \times n$ matrix of functions in $L^1(\R)$ . Write $\hat K(\xi) = \int_{\R} K(x)\, e^{i\xi x}\, dx$ entrywise, the Fourier transform, and let $A(\xi) = I - \hat K(\xi)$ . Assume that $\det A(\xi) \ne 0$ for all $\xi \in \R$ , so that $A(\xi)$ is an invertible matrix for every real $\xi$ . Let $r(\xi) = (\xi - i)/(\xi + i)$ . Then:

There are $n \times n$ matrices $K_+$ and $K_-$ of $L^1(\R)$ functions, with $K_+ = 0$ on $(-\infty, 0)$ and $K_- = 0$ on $(0, \infty)$ . There are integers $\kappa_1 \ge \kappa_2 \ge \cdots \ge \kappa_n$ such that $A(\xi) = \bigl(I - \hat K_-(\xi)\bigr)\, \operatorname{diag}\bigl(r(\xi)^{\kappa_1}, \ldots, r(\xi)^{\kappa_n}\bigr)\, \bigl(I - \hat K_+(\xi)\bigr), \quad \xi \in \R.$ Here $\det(I - \hat K_+)$ has no zeros in the closed upper half-plane and $\det(I - \hat K_-)$ has no zeros in the closed lower half-plane.
The integers $\kappa_1, \ldots, \kappa_n$ , the partial indices, are uniquely determined by $A$ , and their sum is the winding number of $\xi \mapsto \det A(\xi)$ about $0$ .
For $1 \le p < \infty$ , the operator $\varphi \mapsto \varphi(x) - \int_0^\infty K(x - t)\, \varphi(t)\, dt$ on $L^p(0, \infty)^n$ is invertible if and only if $\kappa_1 = \cdots = \kappa_n = 0$ .

Remarks

The factors themselves are not unique, but the partial indices are. For $n = 1$ this is the scalar Wiener–Hopf factorization, with the single index $\kappa_1$ equal to the winding number. The proof cannot take logarithms, because matrices do not commute; one instead reduces to a matrix Riemann–Hilbert problem or works in the algebra of matrix functions with $L^1$ Fourier data.

The total index alone does not decide invertibility. Example: $A = \operatorname{diag}\bigl(r, r^{-1}\bigr)$ has $\det A = 1$ , so the winding number of the determinant is $0$ . Its partial indices are $1$ and $-1$ , so the corresponding system of Wiener–Hopf equations is not invertible. The first equation has a range of codimension $1$ and the second a kernel of dimension $1$ . Each entry is admissible, since $r(\xi) = 1 - \frac{2i}{\xi + i}$ and $\frac{2i}{\xi + i}$ is the Fourier transform of the function equal to $2e^{-x}$ for $x > 0$ and to $0$ for $x \le 0$ . The inverse $r^{-1} = 1 + \frac{2i}{\xi - i}$ is admissible in the same way, with $\frac{2i}{\xi - i}$ the transform of a function supported on $(-\infty, 0]$ .

Going deeper

For a student

The partial indices are the matrix version of the index of a Fredholm operator, but they carry more information than the index. They are unstable: a small perturbation of $A$ can change them, while their sum stays the same. The same factorization governs the matrix Riemann–Hilbert problem on a closed contour, with the circle in place of the real line. In applications, such as diffraction by a wedge or elasticity with mixed conditions, finding the factors explicitly is the main difficulty, and only special classes of matrices admit closed forms.

Reading path

To understand this page from the ground up, read these 42 pages in order. The foundations come first.

Binary operation definition
Complex numbers object
Element definition
Fourier transform concept
Fredholm index concept
Fredholm operator concept
Function concept
Group concept
Holomorphic function concept
Invertible matrix concept
Proposition concept
Real numbers object
System of linear integral equations concept
Wiener–Hopf equation concept
Commutativity definition
Contour integral definition
Set definition
Laplace transform definition
Implication definition
Predicate definition
Cauchy principal value definition
Winding number definition
Ordered pair definition
Heaviside step function object
Abelian group definition
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Dirac delta function object
Circle group object
Subset definition
Uniqueness quantifier definition
Binary relation definition
Wiener's lemma statement
Function definition
Wiener–Hopf factorization statement
Metric space definition
Lipschitz map definition
Hölder continuity definition
Sokhotski–Plemelj theorem statement
Riemann–Hilbert problem problem
Matrix Wiener–Hopf factorization statement

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Matrix Riemann–Hilbert problem

Tag	`2M672PP7QR`
URI	`m2://2M672PP7QR`
Kind	statement, theorem
Writing	W0 stub
MSC	45F15, 47A68, 47A56
Status	This copy of the site has no build data. Run `m2 status 2M672PP7QR`.
About	System of linear integral equations, Wiener–Hopf equation
Uses	Wiener–Hopf factorization, System of linear integral equations, Fourier transform, Invertible matrix, Winding number, Riemann–Hilbert problem, Wiener–Hopf equation

Matrix Wiener–Hopf factorization

Statement

Remarks

Going deeper

For a student

Reading path

Where this leads

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