Inverse Scattering Algorithms for the Commutant Lifting Theorem

Abstract

In this chapter we will obtain a complete characterization of all contractive intertwining liftings B of A in the commutant lifting theorem. As in Section VII.8 we show that there is a one to one correspondence between the set of all contractive intertwining liftings B of A and the set of all choice sequence initiated from G to G′. Then we present an inverse scattering algorithm to compute the choice sequence associated with B. By using an approach based on choice sequences we give a Schur type representation of B. This Schur representation shows that the set of all contraction intertwining liftings B of A is given by a “linear fractional transformation” of a contractive analytic function R in H^∞(G, G′). Next by recursively using Theorem 3.1 in Chapter IV we present four more scattering algorithms for the commutant lifting theorem. We will give another proof of the fact that there is a one to one correspondence between the set of all contractive intertwining liftings of A and the set of all choice sequences initiated from G to G′. If A is a strict contraction, then by choosing the appropriate operators T and T′ two of these algorithms reduce to the Schur algorithm, namely to Procedure 4.3 and Procedure 5.6 used in Chapter Ito compute the reflection coefficients from the Carathéodory data. To complete this chapter we will use our Schur representation to obtain the Adamjan, Arov and Krein formula for characterizing all contractive intertwining liftings of a strictly contractive Hankel operator. This naturally leads to a computational procedure for computing all contractive intertwining liftings for a strictly contractive Hankel operator whose symbol is rational. As a demonstration of this procedure we provide another derivation of the Schur representation (I.3.8) used in solving the Carathéodory interpolation problem.