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Operator Theory and Indefinite Inner Product Spaces pp 147–162Cite as

Spectral Properties of Operator Polynomials with Nonnegative Coefficients

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Part of the book series: Operator Theory: Advances and Applications ((OT,volume 163))

Abstract

We study properties of the polynomial Q(λ) = λmI − S(λ) where S(λ) = λlAl + ...+ λA1 + A0 and 1 ≤ m < l. The coefficients Al,...,A0 are in the positive cone of an ordered Banach algebra or are positive operators on a complex Banach lattice E and I is the identity. We study the properties of the spectral radius of S(λ) if λ is a nonnegative real number, and its connection with the existence of spectral divisors with nonnegative coefficients in the considered sense. We prove factorization results for nonnegative elements in an ordered decomposing Banach algebra with closed normal algebra cone and in the Wiener algebra. Earlier results on monic (nonnegative) operator polynomials are applied to the operator polynomial class studied here.

This work was completed with partial support of the Hungarian National Science Grants OTKA Nos T-030042 and T- 047276 and partial support of the DAAD and the Technical University of Berlin.

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Author information

Authors and Affiliations

  1. Institut für Mathematik, MA 6-4, Technische Universität Berlin Fakultät II, Straße des 17. Juni 136, D-10623, Berlin, Germany

    Karl-Heinz Förster

  2. Department of Analysis Institute of Mathematics, Technical University of Budapest, H-1521, Budapest, Hungary

    Béla Nagy

Authors
  1. Karl-Heinz Förster
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  2. Béla Nagy
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Strathclyde, 26 Richmond Street, Glasgow, G1 1XH, UK

    Matthias Langer

  2. Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstrasse 8-10 / 101, 1040, Wien, Austria

    Annemarie Luger  & Harald Woracek  & 

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Dedicated to Professor Dr. Heinz Langer

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© 2005 Birkhäuser Verlag Basel/Switzerland

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Förster, KH., Nagy, B. (2005). Spectral Properties of Operator Polynomials with Nonnegative Coefficients. In: Langer, M., Luger, A., Woracek, H. (eds) Operator Theory and Indefinite Inner Product Spaces. Operator Theory: Advances and Applications, vol 163. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7516-7_7

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