Existence and Uniqueness of AP Factorization

Abstract

We first prove that if a matrix function has an AP factorization, then this factorization is essentially unique. This allows us to define the almost periodic indices of an almost periodic matrix function and the geometric mean in the case where all almost periodic indices coincide. The question on the existence of an AP factorization is much more complicated. Satisfactory answers will be given for scalar-valued functions and for periodic matrix functions. We also construct an invertible almost periodic matrix polynomial that has no AP factorization. The existence of such matrix polynomials uncovers a big difference between WH and AP factorization. We finally study the question whether the almost periodic indices are stable under small perturbations, after which we can prove that the geometric mean depends continuously on the matrix function.