Bounded indecomposable semigroups of non-negative matrices

Bounded indecomposable semigroups of non-negative matrices A semigroup $${\mathfrak{S}}$$ of non-negative n × n matrices is indecomposable if for every pair i, j ≤ n there exists $${S\in\mathfrak{S}}$$ such that (S) ij ≠ 0. We show that if there is a pair k, l such that $${\{(S)_{kl} : S\in\mathfrak{S}\}}$$ is bounded then, after a simultaneous diagonal similarity, all the entries are in [0, 1]. We also provide quantitative versions of this result, as well as extensions to infinite-dimensional cases. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Bounded indecomposable semigroups of non-negative matrices

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Publisher
SP Birkhäuser Verlag Basel
Copyright
Copyright © 2009 by Birkhäuser Verlag Basel/Switzerland
Subject
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-009-0024-5
Publisher site
See Article on Publisher Site

Abstract

A semigroup $${\mathfrak{S}}$$ of non-negative n × n matrices is indecomposable if for every pair i, j ≤ n there exists $${S\in\mathfrak{S}}$$ such that (S) ij ≠ 0. We show that if there is a pair k, l such that $${\{(S)_{kl} : S\in\mathfrak{S}\}}$$ is bounded then, after a simultaneous diagonal similarity, all the entries are in [0, 1]. We also provide quantitative versions of this result, as well as extensions to infinite-dimensional cases.

Journal

PositivitySpringer Journals

Published: Jun 20, 2009

References

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