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