Minimal and optimal linear discrete time-invariant dissipative scattering systems

For an operator-valued function θ in the Schur class a new geometric proof, using state space considerations only, of the construction of a minimal and optimal realization is given. A minimal and optimal realization also appears as a restricted shift realization where the state space is the completion of the range of the associated Hankel operator in the de Branges-Rovnyak norm associated with θ. It is also shown that minimal and optimal, and minimal and star-optimal realizations of a rational matrix function in the Schur class are intimately connected to the extremal positive solutions of the associated Kalman-Yakubobich-Popov operator inequality.