An operator theoretical approach towards systems over the signal space \(\ell_2(\mathbb Z)\)
Let \(P:D(P)\subset l_2(\mathbb{Z})^m\to l_2(\mathbb{Z})^p\) be a shift-invariant, densely defined linear operator. Then \(P\) is called a linear discrete-time time-invariant system over the signal space \(l_2(\mathbb{Z})\) (shortly, \(P\) is a LTI\(^{p\times m}\)-system). An LTI\(^{p\times m}\)-system \(P\) is called causal if the truncation of its graph \(G(P)\) to each discrete semi-axis \(\{j\in\mathbb{Z}:j\leq n\}\) is a graph of an operator for every \(n\in\mathbb{Z}\). An LTI\(^{p\times m}\)-system \(P\) is called closed (resp., closable) if the operator \(P\) is closed (resp., closable). \textit{T. T. Georgiou} and \textit{M. C. Smith} [IEEE Trans. Autom. Control 40, 516-518 (1995; Zbl 0821.93006)] gave an example of an LTI\(^{1\times 1}\)-system which is causal, closable, but whose closure is not causal. Of course, this phenomenon makes a development of system theory over the signal space \(l_2(\mathbb{Z})\) difficult. In order to overcome this difficulty, the author introduces the notion of causally closable LTI-system (i.e., which is closable and has the causal closure) and gives sufficient conditions for causal closability of a system. She also characterizes the transfer functions of causal, closable LTI\(^{p\times m}\)-systems satisfying a weak extra condition as \(p\times m\) matrices whose entries are quotients of \(H_\infty(\mathbb{D})\)-functions.
scientific article; zbMATH DE number 1971836