An Operator Theoretical Approach Towards Systems over the Signal Space l2 (Z)

In this paper a system is considered as a (possibly unbounded) linear operator from l2 (Z)^{m} to l(2) (Z)^(p). Georgiou and Smith [6] noted that there are intrinsic difficulties in using l(2) (Z) as underlying signal space, since even a simple causal convolution system is not closed and an extended definition of the system is not causal. We discuss these difficulties and we develop necessary and sufficient conditions for notions such as causality, closability and causal closability.