On the Lyapunov equation, coinvariant subspaces and some problems related to spectral factorizations

A geometric approach to stochastic realization theory, and hence to spectral factorization problems, has been developed by Lindquist and Picci (1985, 1991) and Lindquist et al. (1995). Most of this work was done abstractly. Fuhrmann and Gombani (1998) adopted an entirely Hardy space approach to this set of problems, studying the set of rectangular spectral factors of given size for a weakly coercive spectral function. The parametrization of spectral factors in terms of factorizations of related inner functions, as developed in Fuhrmann (1995), had to be generalized. This led to a further understanding of the partial order introduced by Lindquist and Picci in the set of stable spectral factors. In the present paper we study the geometry of finite dimensional coinvariant subspaces of a vectorial Hardy space H-+(2) via realization theory, emphasizing the role of the Lyapunov equation in lifting the Hardy space metric to the state space domain. We follow this by deriving state space formulas for rectangular spectral factors as well as for related inner functions arising in Fuhrmann and Gombani (1998). Finally, we develop a state space approach to the analysis of the partial order of the set of rectangular spectral factors of a given spectral function and its representation in terms of inner functions.