Almost periodic factorization of certain block triangular matrix functions

Let G(x) = [ eIm 0 c−1e−iνx + c0 + c1e e−iλxIm ] , where cj ∈ Cm×m, α, ν > 0 and α+ ν = λ. For rational α/ν such matrices G are periodic, and their Wiener-Hopf factorization with respect to the real line R always exists and can be constructed explicitly. For irrational α/ν, a certain modification (called an almost periodic factorization) can be considered instead. The case of invertible c0 and commuting c1c −1 0 , c−1c −1 0 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 c0 is not invertible but the cj commute pairwise (j = 0,±1). The complete description is obtained when m ≤ 3; for an arbitrary m, certain conditions are imposed on the Jordan structure of cj . 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.