Almost periodic factorization of certain block triangular matrix functions

Abstract:

Let \[ G(x)=\left [\begin {matrix}e^{i\lambda x}I_m & 0\ c_{-1}e^{-i\nu x}+c_0+c_1 e^{i\alpha x} & e^{-i\lambda x}I_m \end {matrix}\right ],\] where $c_j\in \mathbb {C}^{m\times m}$, $\alpha ,\nu >0$ and $\alpha +\nu =\lambda$. For rational $\alpha /\nu$ such matrices $G$ are periodic, and their Wiener-Hopf factorization with respect to the real line $\mathbb {R}$ always exists and can be constructed explicitly. For irrational $\alpha /\nu$, a certain modification (called an almost periodic factorization) can be considered instead. The case of invertible $c_0$ and commuting $c_1c_0^{-1}$, $c_{-1}c_0^{-1}$ was disposed of earlier—it was discovered that an almost periodic factorization of such matrices $G$ does not always exist, and a necessary and sufficient condition for its existence was found. This paper is devoted mostly to the situation when $c_0$ is not invertible but the $c_j$ commute pairwise ($j=0,\pm 1$). The complete description is obtained when $m\leq 3$; for an arbitrary $m$, certain conditions are imposed on the Jordan structure of $c_j$. Difficulties arising for $m=4$ are explained, and a classification of both solved and unsolved cases is given. The main result of the paper (existence criterion) is theoretical; however, a significant part of its proof is a constructive factorization of $G$ in numerous particular cases. These factorizations were obtained using Maple; the code is available from the authors upon request.