Quasi-separable Systems

Patrick Dewilde () and Alle-Jan van der Veen ()
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Patrick Dewilde: Institute for Advanced Study, Technische Universität München
Alle-Jan van der Veen: Delft University of Technology, Signal Processing Systems

Chapter 38 in Operator Theory, 2026, pp 1029-1060 from Springer

Abstract: Abstract Systems of linear equations are called quasi-separable when their input-output operator decomposes in a forward (lower) and a backward (upper) operator, which both have a recursive structure characterized by evolving finite-dimensional state vectors, and linear equations that link the evolving state vector to its next value, called state transition matrices. They generalize the notion of semi-separability in that their characteristic structure is not constrained to pure outer products. This type of operator occurs commonly not only in Dynamical System Theory for discrete-time, time-variant systems with a finite-dimensional state space but also as the result of the discretization of partial differential equations or integral equations. They form a natural generalization of finite matrices. A complete theory based on sequences of state transition matrices is available for them. This 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. Quasi-separable systems depend on a single principal indexing variable, often identified with time. 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. This approach translates to attractive numerical algorithms, such as the celebrated “square-root algorithm,” which uses proven numerically stable operations such as QR-factorization or Singular Value Decomposition (SVD) recursively.

Keywords: Quasi-separability; Inner-outer factorization; LU factorization; Spectral factorization; Time-variant systems; Moore-Penrose inverse (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/978-3-032-16356-1_52

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