The Schur Representation

Abstract

In this chapter we will use Redheffer products to obtain another Schur type representation for the commutant lifting theorem. We begin by presenting some elementary properties of Redheffer products. Then we will use Redheffer products to obtain the characterization of all 2 by 2 matrix contractions in Section 1V.3, solve an operator version of the classical Caratheodory interpolation problem, and obtain a layer peeling algorithm to compute the corresponding choice sequence. Then we will use Redheffer products to obtain the Schur representation for the commutant lifting theorem in Corollary XIII.3.5. The advantage of this approach is that it bypasses the use of choice sequences. From this we will easily obtain another Schur representation for the commutant lifting theorem. This will readily lead to a computational procedure for solving the operator version of the classical Hermite-Fejer interpolation problem and other interpolation problems involving *-stable contractions. A network interpretation of these results will also be given. We will also give another computational procedure for computing the set of all contractive intertwining liftings for a finite rank Hankel operator.