On submultiplicativity of spectral radius and transitivity of semigroups
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- by Heydar Radjavi and Peter Rosenthal PDF
- Proc. Amer. Math. Soc. 135 (2007), 163-168 Request permission
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
- Jakob Levitzki, Über nilpotente Unterringe, Math. Ann. 105 (1931), no. 1, 620–627 (German). MR 1512728, DOI 10.1007/BF01455832
- M. Omladič, H. Radjavi, P. Rosenthal, and A. Sourour, Inequalities for products of spectral radii, Proc. Amer. Math. Soc. 129 (2001), no. 8, 2239–2243. MR 1695115, DOI 10.1090/S0002-9939-01-05500-9
- Heydar Radjavi and Peter Rosenthal, Simultaneous triangularization, Universitext, Springer-Verlag, New York, 2000. MR 1736065, DOI 10.1007/978-1-4612-1200-3
- Yu. V. Turovskii, Volterra semigroups have invariant subspaces, J. Funct. Anal. 162 (1999), no. 2, 313–322. MR 1682061, DOI 10.1006/jfan.1998.3368
Additional Information
- Heydar Radjavi
- Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
- MR Author ID: 143615
- 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
- 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
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 163-168
- MSC (2000): Primary 47D03
- DOI: https://doi.org/10.1090/S0002-9939-06-08446-2
- MathSciNet review: 2280184