Let H, E and E_* denote, respectively, the state space, the input space, and the output space, all of which are assumed to be Hilbert spaces. A unitary colligation is defined as a unitary operator U : H × E →

H × E_*, having the form , and the associated conservative system.

The fact that U is unitary generates the energy balance relations

As a multidimensional analogue of linear system (1) with U unitary, the authors consider three state-space representations for a conservative, multidimensional input-state-output linear system (labeled Roesser, Fornasini-Marchesini, and Kalyuzhniy-Verbovetzky) and sort out the equivalences and connections among them. The main part of this paper is a study of admissible system trajectories (namely, trajectories n →(u(n),x n),y(n) of finite energy, where the variable n varies over the integer lattice Z^d) from a coordinate- free point of view. Although the results are essentially system-theoretic, they have connections with operator theory.

The authors were successful in deriving general energy balance relations, and in proving the equivalence of various expressions for the norm of a finite-energy trajectory. They also show the solvability of the initial value problem associated with any shift-invariant sublattice &OHgr; ⊂ Z^d.

The results are interesting from a theoretical point of view. However, some theorems are rather long (for instance, Theorem 2.4. covers from part of page 155 up to the top part of page 157), which might limit the value of these results in applications. Although the paper is very long and its notations are somewhat cumbersome, it is very well written and documented.