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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On submultiplicativity of spectral radius and transitivity of semigroups


Authors: Heydar Radjavi and Peter Rosenthal
Journal: Proc. Amer. Math. Soc. 135 (2007), 163-168
MSC (2000): Primary 47D03
Published electronically: June 20, 2006
MathSciNet review: 2280184
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Abstract: It is shown that a transitive, closed, homogeneous semigroup of linear transformations on a finite-dimensional space either has zero divisors or is simultaneously similar to a group consisting of scalar multiples of unitary transformations. The proof begins with the result that for each closed homogeneous semigroup with no zero divisors there is a $ k$ such that the spectral radius satisfies $ r(AB) \leq k r(A) r(B)$ for all $ A$ and $ B$ in the semigroup.

It is also shown that the spectral radius is not $ k$-submultiplicative on any transitive semigroup of compact operators.


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Additional Information

Heydar Radjavi
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: hradjavi@cpu105.math.uwaterloo.ca

Peter Rosenthal
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
Email: rosent@math.toronto.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08446-2
PII: S 0002-9939(06)08446-2
Received by editor(s): March 22, 2005
Received by editor(s) in revised form: July 27, 2005
Published electronically: June 20, 2006
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.