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A Panorama of Modern Operator Theory and Related TopicsBanded Matrices, Banded Inverses and Polynomial Representations for Semi-separable Operators

A Panorama of Modern Operator Theory and Related Topics: Banded Matrices, Banded Inverses and... [The paper starts out with exploring properties of the URV factorization in the case of banded matrices or operators with banded inverse, showing that they result in factors with the same properties. Then it gives a derivation of representations for general semi-separable operators (matrices) as ratios of minimally banded matrices. It shows that under pretty general technical conditions (uniform reachability and/or controllability in finite time), left and right polynomial factorizations exist that are unique (canonical) when the factors are properly restrained. Next, it provides Bezout relations for these factors, explicit formulas for all the terms in these relations and an introduction to potential new applications such as Löwner type interpolation theory for (general) matrices.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A Panorama of Modern Operator Theory and Related TopicsBanded Matrices, Banded Inverses and Polynomial Representations for Semi-separable Operators

Part of the Operator Theory: Advances and Applications Book Series (volume 218)
Editors: Dym, Harry; Kaashoek, Marinus A.; Lancaster, Peter; Langer, Heinz; Lerer, Leonid
Dewilde, Patrick
Springer Journals — Jan 3, 2012
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28 pages

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Publisher
Springer Basel
Copyright
© Springer Basel AG 2012
ISBN
978-3-0348-0220-8
Pages
241 –268
DOI
10.1007/978-3-0348-0221-5_10
Publisher site
See Chapter on Publisher Site

Abstract

[The paper starts out with exploring properties of the URV factorization in the case of banded matrices or operators with banded inverse, showing that they result in factors with the same properties. Then it gives a derivation of representations for general semi-separable operators (matrices) as ratios of minimally banded matrices. It shows that under pretty general technical conditions (uniform reachability and/or controllability in finite time), left and right polynomial factorizations exist that are unique (canonical) when the factors are properly restrained. Next, it provides Bezout relations for these factors, explicit formulas for all the terms in these relations and an introduction to potential new applications such as Löwner type interpolation theory for (general) matrices.]

Published: Jan 3, 2012

Keywords: Semi-separable systems; quasi-separable systems; URV decomposition; canonical polynomial forms; Bezout equations; Loewner interpolation; time varying dynamical systems

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