Abstract
In this chapter we consider matrices with operator or matrix entries, often called block matrices. First we obtain a complete characterization of all 2 by 2 contractive block matrices. Then we develop a Levinson type algorithm to determine whether or not a block matrix is contractive. As an application we will present a new method for solving the Carathéodory interpolation problem. Although this chapter is presented in a Hilbert space framework, we recommend to the reader who is unfamiliar with elementary functional analysis to consider all spaces as Euclidian spaces, that is, different Cn spaces with their standard inner product.
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Notes and Comments
Arov, D.Z. and Z. Grossman, Scattering matrices in the theory of dilation of isometric operators, Soviet Math. Doklady, 27 (1983) pp. 518–522.
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.
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Delsarte, P., Genin, Y. and Y. Kamp, Schur parameterization of positive definite block, Toeplitz systems, SIAM J. Appl. Math., 36 (1979) pp. 34–46.
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© 1990 Springer Basel AG
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Foias, C., Frazho, A.E. (1990). Contractive Expansions on Euclidian and Hilbert Space. 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_4
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DOI: https://doi.org/10.1007/978-3-0348-7712-1_4
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-7714-5
Online ISBN: 978-3-0348-7712-1
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