In this chapter we give a simple and straightforward treatment of the spectral factorization problem of a positive operator Ω∈X into Ω = W*W, where Wi ∈ U is outer. We only consider the case where Ω is a strictly positive operator and where its causal part is bounded and has a u.e. stable realization. This leads to a recursive Riccati equation with time-varying coefficients for which the minimal positive definite solution leads to the outer factor. The theory also includes a formulation of a time-varying (strictly-) positive real lemma. In addition, we provide connections with related problems discussed in previous chapters in which Riccati equations appear as well, such as inner-outer factorization and orthogonal embedding. The results can no doubt be formulated in a more general way where strict positivity is not assumed, but we consider these extensions as laying outside the scope of the book.
KeywordsRiccati Equation Lyapunov Equation Embedding Problem Spectral Factorization Nest Algebra
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