Semi- and Quasi-separable Systems

The main objects of this chapter are “semi-separable systems,” sometimes called “quasi-separable systems.” These are systems of equations, in which the operator has a special structure, called “semi-separable” in this chapter. By this is meant that the operator, although typically infinite dimensional, has a recursive structure determined by sequences of finite matrices, called transition matrices. This type of operator occurs commonly in Dynamical System Theory for systems with a finite dimensional state space and/or in systems that arise from discretization of continuous time and space. They form a natural generalization of finite matrices and a complete theory based on sequences of finite matrices is available for them. The chapter concentrates on the invertibility of such systems: either the computation of inverses when they exist, or the computation of approximate inverses of the Moore–Penrose type when not. Semi-separable systems depend on a single principal variable (often identified with time or a single dimension in space). Although there are several types of semi-separable systems depending on the continuity of that principal variable, the present chapter concentrates on indexed systems (so-called discrete-time systems). This is the most straightforward and most appealing type for an introductory text. The main workhorse is “inner–outer factorization,” a technique that goes back to Hardy space theory and generalizes to any context of nest algebras, as is the one considered here. It is based on the definition of appropriate invariant subspaces in the range and co-range of the operator. It translates to attractive numerical algorithms, such as the celebrated “square-root algorithm,” which uses proven numerically stable operations such as QR-factorization and singular value decomposition (SVD).