A time-variant norm constrained interpolation problem arising from relaxed commutant lifting

TL;DR

This paper introduces and analyzes a time-variant interpolation problem related to relaxed commutant lifting, providing explicit solutions and extending classical results to a dynamic setting.

Contribution

It develops a time-variant analogue of the relaxed commutant lifting problem and explicitly characterizes all solutions in this new context.

Findings

Explicit description of all solutions to the time-variant interpolation problem

Extension of classical theorems to a dynamic, time-variant setting

Connection to the analysis of the three chain completion problem

Abstract

A time-variant analogue of an interpolation problem equivalent to the relaxed commutant lifting problem is introduced and studied. In a somewhat less general form the problem already appears in the analysis of the set of all solutions to the three chain completion problem. The interpolants are upper triangular operator matrices of which the columns induce contractive operators. The set of all solutions of the problem is described explicitly. The results presented are time-variant analogues of the main theorems in [A.E. Frazho, S. ter Horst, and M.A. Kaashoek, All solutions to the relaxed commutant lifting problem, Acta Sci. Math. (Szeged) 72 (2006), 299--318].