Wiener–Hopf factorization for a group of exponentials of nilpotent matrices

A complete study of the generalized factorization for a group of 2×2 matrix functions of the form G=I+γN, where , I denotes the 2×2 identity matrix and N represents a rational nilpotent matrix function, is presented. A closely related class involving the same matrix N is also studied. The canonical and non-canonical factorizations are considered and explicit formulas are obtained for the partial indices and the factors in such factorizations. It is shown in particular that only one of the columns in the factors needs to be determined, as a solution to a homogeneous linear Riemann–Hilbert problem, the other column being expressed in terms of the first. Necessary and sufficient conditions for existence of a canonical factorization within the same class are established, as well as explicit formulas for the factors in this case.