All solutions to an operator nevanlinna-pick interpolation problem

A. E. Frazho, S. ter Horst*, M. A. Kaashoek

*Corresponding author for this work

Research output: Chapter in Book / Report / Conference proceedingConference contributionAcademicpeer-review

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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.

Original languageEnglish
Title of host publicationOperator Theory in Different Settings and Related Applications
Subtitle of host publication26th IWOTA, Tbilisi, July 2015
EditorsRoland Duduchava, Marinus A. Kaashoek, Nikolai Vasilevski, Victor Vinnikov
PublisherSpringer International Publishing Switzerland
Pages139-220
Number of pages82
ISBN (Electronic)9783319625270
ISBN (Print)9783319625263, 9783030096755
DOIs
Publication statusPublished - 2018

Publication series

NameOperator Theory: Advances and Applications
Volume262
ISSN (Print)0255-0156
ISSN (Electronic)2296-4878

Funding

The present work is based on the research supported in part by the National Research Foundation of South Africa. Any opinion, finding and conclusion or recommendation expressed in this material is that of the authors and the NRF does not accept any liability in this regard.

FundersFunder number
National Research Foundation
National Research Foundation90670, 93406

    Keywords

    • Co-isometric systems
    • Entropy
    • Linear fractional transformations
    • Nevanlinna-pick interpolation
    • Operator optimisation problems

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