Normalization of Wiener–Hopf factorization for $2\times 2$ matrix functions and its application

In this work we cover a gap existing in the general Wiener-Hopf factorization theory of matrix functions. It is known that factors in such factorization are not determined uniquely and in the general case, there are no known ways of normalizing the factorization ensuring its uniqueness. In the work we introduce the notion of $P$-normalized factorization. Such normalization ensures the uniqueness of the Wiener-Hopf factorization and gives an opportunity to find the Birkhoff factorization. For the second order matrix function we show that the factorization of each matrix function can be reduced to the $P$-normalized factorization. We describe all possible types of such factorizations, obtain the conditions ensuring the existence of such normalization and find the form of the factors for such type of the normalization. We study the stability of $P$-normalization under a small perturbation of the initial matrix function. The results are applied for specifying the Shubin theorem on the continuity of the factors and for obtaining the explicit estimates of the absolute errors of the factors for an approximate factorization.