On the Hankel-norm Approximation of Upper-triangular Operators and Matrices

A matrix T = Tij ∞ i,j=−∞, which consists of a doubly indexed collection fT ijg of operators, is said to be upper when Tij = 0 for i > j. We consider the case where the Tij are finite matrices and the operator T is bounded, and such that the Tij are generated by a strictly stable, non-stationary but linear dynamical state space model or colligation. For such a model, we consider model reduction, which is a procedure to obtain optimal approximating models of lower system order. Our approximation theory uses a norm which generalizes the Hankel norm of classical stationary linear dynamical systems. We obtain a parametrization of all solutions of the model order reduction problem in terms of a fractional representation based on a non-stationary J-unitary operator constructed from the data. In addition, we derive a state space model for the so-called maximum entropy approximant. In the stationary case, the problem was solved by Adamyan, Arov and Krein in their paper on Schur-Takagi interpolation. Our approach extends that theory to cover general, non-Toeplitz upper operators.