---
tag: 2M672PP7QR
kind: statement
role: theorem
name: Matrix Wiener–Hopf factorization
msc: [45F15, 47A68, 47A56]
uses: [BYKP1V2AQN, G9WR1GDE7X, DPVM34ADZV, 0EB9N0S33C, 0X7SBS8E0F, RY8YQ9JZF9, QYY4NMDY5H]
about: [G9WR1GDE7X, QYY4NMDY5H]
sources:
  - url: https://en.wikipedia.org/wiki/Wiener%E2%80%93Hopf_method
---

## 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|DPVM34ADZV]], 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|0EB9N0S33C]] for every real $\xi$.
Let $r(\xi) = (\xi - i)/(\xi + i)$.
Then:

1. 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.
2. The integers $\kappa_1, \ldots, \kappa_n$, the *partial indices*, are uniquely determined by $A$, and their sum is the [[winding number|0X7SBS8E0F]] of $\xi \mapsto \det A(\xi)$ about $0$.
3. 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|BYKP1V2AQN]], 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|QYY4NMDY5H]] 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 $(-\infty, 0]$.

## Explanations

### L1 human

A system of Wiener–Hopf equations has a matrix symbol instead of a scalar one.
The symbol can still be split into a forward-looking part, a backward-looking part and a simple diagonal middle part.
The middle part is described by a list of whole numbers, the partial indices, and these decide whether the system has exactly one solution.
Unlike the scalar case, the winding number of the determinant is not enough: the indices can cancel in the sum.

### L2 human

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|RY8YQ9JZF9]] 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.