Non-linear minimal square spectral factorization

Recent work has identified a correspondence between minimal square spectral factors of a given spectral density, invariant subspaces of an associated Hamiltonian matrix, solutions of an appropriate algebraic Riccati equation and minimal unitary left divisors of a certain unitary function on the imaginary line. The purpose of this report is to obtain a natural extension of these ideas to a non-linear setting, at least for the special case where only minimal square stable spectral factors are considered.