LU Factorizations for ℕ × ℕ-Matrices and Solutions of the k[S]-Hierarchy and Its Strict Version

Let S be the N×N-matrix of the shift operator and let k denote the field of real or complex numbers. We consider two different deformations of the commutative algebra k[S], together with the evolution equations of the deformations of the powers {Si,i⩾1}. They are called the k[S]-hierarchy and the strict k[S]-hierarchy. For suitable Banach spaces B, we explain how LU factorizations in GL(B) can be used to produce dressing matrices of both hierarchies. These dressing matrices correspond to bounded operators on B, a class far more general than the one used at a prior construction. This wider class of solutions of both hierarchies makes it possible to treat reductions of both systems. The matrix coefficients of these matrices are shown to be quotients of analytic functions. Moreover, we present a subgroup Gcpt(B) of GL(B) such that the procedure with LU factorizations works for each g∈Gcpt(B).