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

A geometric approach to stochastic realization theory, and hence to spectra l factorization problems, has been developed by Lindquist and Picci (1985, 1991) and Lindquist et al. (1995). Most of this work was done abstractly. F uhrmann 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 spect ral factors in terms of factorizations of related inner functions, as devel oped in Fuhrmann (1995), had to be generalized. This led to a further under standing 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 coinvarian t subspaces of a vectorial Hardy space H-+(2) via realization theory, empha sizing 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 ari sing in Fuhrmann and Gombani (1998). Finally, we develop a state space appr oach to the analysis of the partial order of the set of rectangular spectra l factors of a given spectral function and its representation in terms of i nner functions.