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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Notes And Comments
Davis, C., A factorization of an arbitrary m x n contractive operator, Proceedings of the Toeplitz Centennial Conference, Tel-Aviv, Birkhäuser-Verlag (1982) pp. 217–232.
Davis, C., Completing a matrix so as to minimize the rank, Topics in Operator Theory and Interpolation; Operator Theory: Advances and Aplications, 29, Ed. I. Gohberg (1988) pp. 87–95.
Johnson, C.R. and L. Rodman, Inertia possibilities for completions of partial Hermitian matrices, Linear and Multilinear Algebra, 16 (1984) pp. 179–195.
Arsene, Gr., Ceausescu, Z. and C. Foias, On intertwining dilations VII, Proc. Coll. Complex Analysis, Joensuu, Lecture Notes in Math., 747 (1979) pp. 24–45.
Frazho, A.E., Three inverse scattering algorithms for the lifting theorem, I. Schur Methods in Operator Theory and Signal Processing; Operator Theory: Advances and Applications, 18, Ed. I. Gohberg (1986) pp. 219–248.
Constantinescu, T., A maximum entropy principle for contractive intertwining dilations, Operator theory: Advances and Applications, 24 (1987) pp. 69–85.
Gohberg, I., Kaashoek, M.A., and H.J. Woerdeman, The band method for positive and strictly contractive extension problems: an alternative version and new applications, Integral Equations and Operator Theory, 12 (1989) pp. 343–382.
Arocena, R., The Gohberg Anniversary Collection; Operator Theory: Advances and Applications, 41, Ed H. Dym, S. Goldberg, M. A. Kaashoek and P. Lancaster (1989) pp. 13–23.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer Basel AG
About this chapter
Cite this chapter
Foias, C., Frazho, A.E. (1990). Inverse Scattering Algorithms for the Commutant Lifting Theorem. In: The Commutant Lifting Approach to Interpolation Problems. OT 44 Operator Theory: Advances and Applications, vol 44. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7712-1_13
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7712-1_13
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-7714-5
Online ISBN: 978-3-0348-7712-1
eBook Packages: Springer Book Archive