Zeros of Spectral Factors, the Geometry of Splitting Subspaces and the Algebraic Riccati Inequality

In this paper it is shown how the zero dynamics of (not necessarily square) spectral factors relate to the splitting subspace geometry of stationary stochastic models and to the corresponding algebraic Riccati inequality. The notion of output-induced subspace of a minimal Markovian splitting subspace, which is the stochastic analogue of the supremal output-pulling subspace in geometric control theory, is introduced. Through this concept, the analysis can be made coordinate-free and straightforward geometric methods can be applied. It is shown how the zero structure of the family of spectral factors relates to the geometric structure of the family of minimal Markovian splitting subspaces in the sense that the relationship between the zeros of different spectral factors is reflected in the partial ordering of minimal splitting subspaces. Finally, the well-known characterization of the solutions of the algebraic Riccati equation is generalized in terms of Lagrangian subspaces invariant under the corresponding Hamiltonian to the larger solution set of the algebraic Riccati inequality.