Abstract
In this chapter we will show how one can use the commutant lifting theorem to solve several classical interpolation problems. Indeed we will treat the Carathéodory interpolation problem, the Nevanlinna-Pick interpolation problem, a Hankel matrix interpolation problem, an extension problem involving contractions intertwining isometries and a Toeplitz inversion problem. We will also use the commutant lifting theorem to obtain some Corona theorems for analytic Toeplitz operators.
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Notes and Comments
Helson, H. and D. Lowdenslager, Prediction theory and Fourier series in several variables, Acta Math., 99 (1958) pp. 165–202.
Helton, J.W., Operator Theory, Analytic Functions, Matrices, and Electrical Engineering, CBMS Regional Conference Series in Math., 68, Amer. Math. Soc., Providence, Rhode Island, 1987.
Hille, E., Functional analysis and semi-groups, American Mathematical Society Providence, Rhode Island, 1948.
Julia, G., Sur les projections des systèmes orthonormaux de l’espace hilbertien, C. R. Acad. Sci. Paris, 218 (1944) pp. 892–895.
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© 1990 Springer Basel AG
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Foias, C., Frazho, A.E. (1990). Geometric Applications of 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_8
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DOI: https://doi.org/10.1007/978-3-0348-7712-1_8
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-7714-5
Online ISBN: 978-3-0348-7712-1
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