Conservative input-state-output systems with evolution on a multidimensional integer lattice

A fundamental object of study in both operator theory and system theory is a discrete-time conservative system (variously also referred to as a unitary system or unitary colligation). In this paper we introduce three equivalent multidimensional analogues of a unitary system where the "time axis" ℤ ^dd>1 is multidimensional. These multidimensional formalisms are associated with the names of Roesser Fornasini and Marchesini and Kalyuzhniy-Verbovetzky. We indicate explicitly how these three formalisms generate the same behaviors. In addition we show how the initial-value problem (including the possibility of "initial conditions at infinity") can be solved for such systems with respect to an arbitrary shift-invariant sublattice as the analogue of the positive-time axis. Some of our results are new even for the d=1 case.