Kernel, Image and Scattering Representations of Passive State/Signal Systems

In this work the characteristic properties of image and kernel representations of passive and conservative state/signal systems are presented. Earlier these representations were introduced in joint papers with Olof J. Staffans on theory of linear state/signal systems for much wider class, namely closed state/signal systems. In the case of passive and conservative state/signal systems the central role in our theory play scattering representations instead of these representations. In this paper the connections between image and scattering representations of a passive state/signal system are established, too. Main notions and results of passive s/s theory are connected with known notions and results from Krein spaces that are intensively used here. At the end an example of passive and conservative state/signal system is demonstrate on a simple quantum graph.