Perron matrix semigroups
Abstract
We consider multiplicative semigroups of real dxd matrices. A semigroup S is called Perron if each of its matrices has a Perron eigenvalue, i.e., an eigenvalue equal to the spectral radius. If all matrices of S leave a proper convex cone invariant, then S is Perron. Our main result asserts the converse: every irreducible Perron semigroup possesses a common invariant cone, provided that some mild assumptions are satisfied. This gives conditions for a set of matrices to share a common invariant cone, which is an important property widely studied in the literature. Then we address the problem to characterize the exceptions, when a Perron semigroup does not have an invariant cone. For d\le 4, all Perron semigroups are classified. For higher dimensions~, several classes of such semigroups are found.
Cite
@article{arxiv.2502.10571,
title = {Perron matrix semigroups},
author = {Vladimir Yu. Protasov},
journal= {arXiv preprint arXiv:2502.10571},
year = {2026}
}