Sets of matrices with singleton spectra generated by positive matrices

Abstract

In this paper we present order versions of Kaplansky's results on triangularizability of semigroups and Lie sets of linear operators with singleton spectra. The Jordan set case is more difficult to deal with. For the Jordan set J of positive matrices with singleton spectra we prove the following:(a)If J contains a nilpotent matrix N with N2≠0, then J is decomposable.(b)If J does not contain nilpotent matrices or if J consists of nilpotent matrices, then J is completely decomposable. We also give an example of an irreducible Jordan set of positive matrices with singleton spectra which contains a nonzero square-zero matrix. Furthermore, we determine the lower bound for the number of standard subspaces invariant under a given Jordan set of positive matrices with singleton spectra which contains a nilpotent matrix.