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



Wiener-Hopf factorization of matrices
Pointers to the literature on matrix-Wiener-Hopf factorization can be found in [A brief historical perspective of the Wiener-Hopf technique:](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.426.46&rep=rep1&type=pdf)



> 
> 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](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.426.9671&rep=rep1&type=pdf) (and this related [note):](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.426.5656&rep=rep1&type=pdf)



> 
> 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.


