Factorization of almost periodic matrix functions (Q1900649)

Let \(AP\) be the space of Bohr almost periodic functions \(f(x) \sim \sum_j f_j e^{i \lambda_j x}\). Let \(\widehat f (\lambda) = M(e^{- i \lambda x} f(x))\) and \(\Omega (f) = \{\lambda \in \mathbb{R} : \widehat f (\lambda) \neq 0\}\), \(\mathbb{R}_\pm = \{x \in \mathbb{R} : \pm x \geq 0\}\), \(AP^\pm = \{f \in AP : \Omega (f) \subset\mathbb{R}_\pm\}\). An \(AP\)-factorization of an \(n \times n\) matrix function \(G\) is given by an identity \(G=G^+\Lambda G^-\), where \((G^+)^{\pm 1}\in AP^+\), \((G^-)^{\pm 1} \in AP^-\) and \(\Lambda (x) = \text{diag} [e^{i \lambda_1 x}, \dots, e^{i \lambda_n x}]\). There are given conditions under which the following matrices are \(AP\)-factorizable: \[ G(x) = \left[ \begin{matrix} e^{i \lambda x} I_m, & 0 \\ - Ae^{i \mu x} + Be^{i \alpha x}, & e^{- i \lambda x} I_m \end{matrix} \right], \quad G(x) = \left[ \begin{matrix} e^{i \lambda x} I_m, & 0 \\ c_{-1} e^{-i \nu x} - c_0 + c_1 e^{i \alpha x}, & e^{-i \lambda x} I_m \end{matrix} \right]. \]