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The Inverse Commutant Lifting Problem. I: Coordinate-Free Formalism

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Abstract

It is known that the set of all solutions of a commutant lifting and other interpolation problems admits a Redheffer linear-fractional parametrization. The method of unitary coupling identifies solutions of the lifting problem with minimal unitary extensions of a partially defined isometry constructed explicitly from the problem data. A special role is played by a particular unitary extension, called the central or universal unitary extension. The coefficient matrix for the Redheffer linear-fractional map has a simple expression in terms of the universal unitary extension. The universal unitary extension can be seen as a unitary coupling of four unitary operators (two bilateral shift operators together with two unitary operators coming from the problem data) which has special geometric structure. We use this special geometric structure to obtain an inverse theorem (Theorem 8.4) which characterizes the coefficient matrices for a Redheffer linear-fractional map arising in this way from a lifting problem. The main tool is the formalism of unitary scattering systems developed in Boiko et al. (Operator theory, system theory and related topics (Beer-Sheva/Rehovot 1997), pp. 89–138, 2001) and Kheifets (Interpolation theory, systems theory and related topics, pp. 287–317, 2002)

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Authors and Affiliations

  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061, USA

    Joseph A. Ball

  2. Department of Mathematics, University of Massachusetts Lowell, Lowell, MA, 01854, USA

    Alexander Kheifets

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  1. Joseph A. Ball
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  2. Alexander Kheifets
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Correspondence to Alexander Kheifets.

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The work of A. Kheifets was partially supported by the University of Massachusetts Lowell Research and Scholarship Grant, project number: H50090000000010.

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Ball, J.A., Kheifets, A. The Inverse Commutant Lifting Problem. I: Coordinate-Free Formalism. Integr. Equ. Oper. Theory 70, 17–62 (2011). https://doi.org/10.1007/s00020-011-1873-4

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