Perron matrix semigroups

Abstract: We consider multiplicative semigroups of real dxd matrices. A semigroup is called Perron if each of its matrices has a Perron eigenvalue, i.e., an eigenvalue equal to the spectral radius. By the Krein-Rutman theorem, a matrix leaving some cone invariant possesses a Perron eigenvalue. Therefore, if all matrices of a semigroup S share a common invariant cone, then S is Perron. Our main result asserts that the converse is true, provided that some mild assumptions are satisfied. Those assumptions cannot be relaxed. This gives conditions for an arbitrary set of matrices to possess a common invariant cone.