Extensions of Perron–Frobenius theory

Extensions of Perron–Frobenius theory The classical Perron–Frobenius theory asserts that, for two matrices $$A$$ and $$B$$ , if $$0\le B \le A$$ and $$r(A)=r(B)$$ with $$A$$ being irreducible, then $$A=B$$ . It has been extended to infinite-dimensional Banach lattices under certain additional conditions, including that $$r(A)$$ is a pole of the resolvent of $$A$$ . In this paper, we prove that the same result holds if $$B$$ is irreducible and $$r(B)$$ is a pole of the resolvent for $$B$$ . We also prove some other interesting extensions of the theorem for infinite-dimensional Banach lattices. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Extensions of Perron–Frobenius theory

Positivity , Volume 17 (4) – Nov 28, 2012
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Publisher
Springer Basel
Copyright
Copyright © 2012 by Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-012-0215-3
Publisher site
See Article on Publisher Site

Abstract

The classical Perron–Frobenius theory asserts that, for two matrices $$A$$ and $$B$$ , if $$0\le B \le A$$ and $$r(A)=r(B)$$ with $$A$$ being irreducible, then $$A=B$$ . It has been extended to infinite-dimensional Banach lattices under certain additional conditions, including that $$r(A)$$ is a pole of the resolvent of $$A$$ . In this paper, we prove that the same result holds if $$B$$ is irreducible and $$r(B)$$ is a pole of the resolvent for $$B$$ . We also prove some other interesting extensions of the theorem for infinite-dimensional Banach lattices.

Journal

PositivitySpringer Journals

Published: Nov 28, 2012

References

  • The irreducibility in ordered Banach algebras
    Alekhno, EA

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