Optimal time-variant systems and factorization of operators. I: Minimal and optimal systems (Q1264490)

The notions of optimal, and star-optimal dissipative scattering (contractive) time-invariant systems were introduced by the first author [J. Oper. Theory 2, 95-126 (1979; Zbl 0461.47005)]. The investigation of such systems was continued in the paper [\textit{D. Z. Arov, M. A. Kaashoek} and \textit{D. R. Pik}, Integral Equations Oper. Theory 29, No. 2, 127-154 (1997; Zbl 0898.47005)]. In particular, two constructions both for a minimal and optimal, and for a minimal and star-optimal contractive system (realizations) were given for prescribed Schur-class (contractive analytic) operator-valued functions as their transfer functions. In the paper under review the authors introduce the notions of optimal, and star-optimal contractive discrete time-variant systems and extend their above-mentioned results to the time-variant setting. Instead of a Schur-class operator-valued function, a contractive block lower triangular operator acting between appropriate Hilbert space vector-valued \(\ell^2\) spaces is considered, and plays a role of the input-output map of a contractive time-variant system. Two different ways of construction both for a minimal and optimal, and for a minimal and star-optimal realization of a prescribed block-triangular contraction are described.