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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 | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. J. Levitzki, Uber nilpotente Unterringe, Math. Ann. 105 (1931), 620-627. MR 1512728
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  • 4. Y.V. Turovski, Volterra semigroups have invariant subspaces, J. Funct. Anal. 162 (1999), 313-322. MR 1682061 (2000d:47017)

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Additional Information

Heydar Radjavi
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Peter Rosenthal
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3

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.

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