A simultaneous Wielandt positivity theorem

A simultaneous Wielandt positivity theorem We consider matrix semigroups $$\mathcal {S}$$ S which are closed under multiplication by complex scalars, and whose norm closure contains no zero-divisors. We show that when every non-zero $$S$$ S in $$\mathcal {S}$$ S is indecomposable and the spectral radius of $$S$$ S is equal to the spectral radius of $$|S|$$ | S | for all $$S$$ S in $$\mathcal {S}$$ S , then $$\mathcal {S}$$ S is effectively positive, in the sense that there exists a diagonal unitary matrix $$D$$ D so that for each $$S$$ S in $$\mathcal {S}$$ S , $$S=\alpha _S D |S| D^{-1}$$ S = α S D | S | D - 1 for some $$\alpha _S \in \mathbb {T}$$ α S ∈ T . We also show the same conclusion holds even if individual indecomposability is weakened to indecomposability of the semigroup as a whole, as long as the semigroup is convex. We give examples showing that all hypotheses are required. We also extend some of these results to compact operators, under additional conditions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

A simultaneous Wielandt positivity theorem

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Publisher
Springer Basel
Copyright
Copyright © 2014 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/s11117-014-0289-1
Publisher site
See Article on Publisher Site

Abstract

We consider matrix semigroups $$\mathcal {S}$$ S which are closed under multiplication by complex scalars, and whose norm closure contains no zero-divisors. We show that when every non-zero $$S$$ S in $$\mathcal {S}$$ S is indecomposable and the spectral radius of $$S$$ S is equal to the spectral radius of $$|S|$$ | S | for all $$S$$ S in $$\mathcal {S}$$ S , then $$\mathcal {S}$$ S is effectively positive, in the sense that there exists a diagonal unitary matrix $$D$$ D so that for each $$S$$ S in $$\mathcal {S}$$ S , $$S=\alpha _S D |S| D^{-1}$$ S = α S D | S | D - 1 for some $$\alpha _S \in \mathbb {T}$$ α S ∈ T . We also show the same conclusion holds even if individual indecomposability is weakened to indecomposability of the semigroup as a whole, as long as the semigroup is convex. We give examples showing that all hypotheses are required. We also extend some of these results to compact operators, under additional conditions.

Journal

PositivitySpringer Journals

Published: May 25, 2014

References

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