Graphs,closability, and causality of linear time-invariant discrete-time systems

In this paper a linear, time-invariant, discrete-time system is considered by regarding it as an (unbounded) operator on 2(N), 2(Z), CN or CZ. The graphs of closed systems are characterized and properties such as causality and closability of systems are studied. It is shown that the closure of a causal, closable system on 2(N), CN and CZ is again causal, whereas this is in general not the case for systems on 2(Z). Necessary and sufficient conditions which guarantee that this property hold for systems on 2(Z) are given. Many of the results extend without change to the continuous-time case.