On the asymptotic regularity of a family of matrices

In this paper we consider bounded families F of complex n 7 n matrices. We give sufficient conditions under which the sequence {r_k(F)^(1/k)}, where r_k(F) is the supremum of the spectral radii of all possible products of k matrices chosen in F, is convergent to its supremum r(F), the so-called (generalized) spectral radius of F. We also illustrate a possible practical application. The research that led to the present paper was partially supported by a grant of the group GNCS of INdAM