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All Solutions to an Operator Nevanlinna–Pick Interpolation Problem

  • A. E. Frazho
  • S. ter Horst
  • M. A. Kaashoek
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 262)

Abstract

The main results presented in this paper provide a complete and explicit description of all solutions to the left tangential operator Nevanlinna– Pick interpolation problem assuming the associated Pick operator is strictly positive. The complexity of the solutions is similar to that found in descriptions of the sub–optimal Nehari problem and variations on the Nevanlinna– Pick interpolation problem in the Wiener class that have been obtained through the band method. The main techniques used to derive the formulas are based on the theory of co-isometric realizations, and use the Douglas factorization lemma and state space calculations. A new feature is that we do not assume an additional stability assumption on our data, which allows us to view the Leech problem and a large class of commutant lifting problems as special cases. Although the paper has partly the character of a survey article, all results are proved in detail and some background material has been added to make the paper accessible to a large audience including engineers.

Keywords

Nevanlinna–Pick interpolation linear fractional transformations co-isometric systems operator optimisation problems entropy 

Mathematics Subject Classification (2010)

Primary 47A57 Secondary 47A48 47A56 47A62 28D20 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Aeronautics and AstronauticsPurdue UniversityWest LafayetteUSA
  2. 2.Department of Mathematics Unit for BMINorth-West UniversityPotchefstroomSouth Africa
  3. 3.Department of MathematicsVU University AmsterdamAmsterdamThe Netherlands

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