Characterization of minimal spectral factors via homeomorphic maps

In this paper a new characterization of the class of all minimal square spectral factors of a given rational spectral density is presented. This characterization consists of two bijective maps which relate the set of minimal square spectral factors to a set of invariant subspaces of a certain matrix and to a set of symmetric solutions of an algebraic Riccati equation. In the second part of the paper it is proven that these two maps are homeomorphisms. This result extends to spectral factorization theory the results of Wimmer (1995), where it is proven that the well known relation between solutions of ARE and invariant subspaces of a certain matrix is, in fact, a homeomorphism