Abstract
If a nonzero linear functional has finite, countable, or bounded range when restricted to an irreducible semigroup \({\mathcal{S}}\) of complex matrices, it is shown that \({\mathcal{S}}\) itself has the same property. Similar results are proven under the hypothesis that a nontrivial ideal of \({\mathcal{S}}\) is finite, countable, or bounded.
Similar content being viewed by others
References
Fillmore, P.A.: On similarity and the diagonal of a matrix. Am. Math. Mon. 76, 167–169 (1969)
Okninski, J.: Semigroups of Matrices. World Scientific, Singapore (1998)
Platonov, V.P., Zaleskii, A.E.: The Auerbach problem. Dokl. Akad. Nauk BSSR 10, 5–6 (1966)
Radjavi, H., Rosenthal, P.: Simultaneous Triangularization. Springer, New York (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jerome A. Goldstein
Rights and permissions
About this article
Cite this article
Radjavi, H., Rosenthal, P. Limitations on the size of semigroups of matrices. Semigroup Forum 76, 25–31 (2008). https://doi.org/10.1007/s00233-007-9004-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-007-9004-x