Almost periodic factorization of certain block triangular matrix functions

Let G(x) = [c(-1)e(-ivxei lambda xIm) + c(0) + c(1)e(iax) e(-i lambda x0)I(m)], where c(j) is an element of C-mxm, alpha, nu > 0 and alpha + nu = lambda. F or rational alpha/nu such matrices G are periodic, and their Wiener-Hopf fa ctorization with respect to the real line R always exists and can be constr ucted explicitly. For irrational alpha/nu, a certain modification (called a n almost periodic factorization) can be considered instead. The case of inv ertible c(0) and commuting c(1)c(0)(-1),c(-1)c(0)(-1) was disposed of earli er-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 co is not inversible but the c(j) commute pairwise (j = 0, +/-1). The complete description is obtai ned when m less than or equal to 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 g iven. The main result of the paper (existence criterion) is theoretical; however, a significant part of its proof is a constructive factorisation of G in nu merous particular cases. These factorizations were obtained using Maple; th e code is available from the authors upon request.