# 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

Positivity, Volume 19 (1) – May 25, 2014
12 pages

/lp/springer_journal/a-simultaneous-wielandt-positivity-theorem-3fL4vzf4jW
Publisher
Springer Journals
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

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