Wiener-Hopf factorization of matrices

Question

Given a $2\times2$ matrix, which entries are functions in the complex plane $$\hat{A}(z)=\left(\begin{array}{cc}a(z)&b(z)\\c(z)&d(z)\end{array}\right)$$Where $a(z),b(z),c(z)$ and $d(z)$ are functions in the complex plane. I would like to obtain two new matrices, denoted as $\hat{\phi}_+(z)$ and $\hat{\phi}_-(z)$, such $\hat{A}(z)=\hat{\phi}_+(z)\hat{\phi}_-(z)$. The meaning of the subindexes is related to the discrete Fourier transform. If we take $\hat{\phi}_+(z)$ $$\int dz e^{i k z}\hat{\phi}_+(z)=0,\;\;\forall\;k<0$$ Similar definitions can be given for $\hat{\phi}_-(z)$ with $k>0$. It means that the Fourier transform of the two functions is only non-zero in a half complex plane.

As far as I know, this is called Wiener-Hopf factorization. My questions are about the conditions that ensure the existence of such a factorization, and how this factorization can be determined (references or examples may help at this point).

Thank you for any help!

Note: Suppose that the entries of $A(z)$ are continium and smooth, and the matrix is invertible a.e. ($\mbox{det}[\hat{A}(z)]\neq0,\forall z\in \mathbb{C}$) By the moment I don't care about pathological cases

Answer 1

Pointers to the literature on matrix-Wiener-Hopf factorization can be found in A brief historical perspective of the Wiener-Hopf technique:

Matrix Wiener-Hopf kernels are fundamentally distinct from their scalar counterparts in that there is no algorithmic approach to determining the factorisation of the transformed kernel. Exact factorisation can be achieved for matrices with certain special features: those that are upper (or lower) triangular in form; those that are of Khrapkov-Daniele, i.e. commutative, form; those whose elements comprise meromorphic functions; kernels with special singularity structure that allows the Wiener-Hopf equation to be recast into uncoupled Riemann-Hilbert problems and $N \times N$ matrices with special algebraic or group structure.

More specifically, for the $2\times 2$ case, see for example Matrix Wiener-Hopf factorization (and this related note):

A method is described for effecting the explicit Wiener-Hopf factorisation of a class of $2\times 2$ matrices. The class is determined such that the factorisation problem can be reduced to a matrix Hilbert problem which involves an upper or lower triangular matrix. Then the matrix Hilbert problem can be further reduced to three scalar Hilbert problems on a half-line, which are solvable in the standard manner.

This explicit method works if $$b\left[a-d\pm\sqrt{(a-d)^2+4bc}\right]^{-1}$$ can be expressed as a ratio of polynomials in $z$. It seems a completely general explicit construction does not exist, not even for $2\times 2$ matrices.