A gap result for the norms of semigroups of matrices

Let ∥·∥ be a matrix norm on Md(C) and let A be a finite set of matrices in Md(C). We define mn(A) to be the maximum norm of a product of n elements of A. We show that there is a gap in the possible growth of mn(A), showing that mn(A) grows either at least exponentially or is bounded by a polynomial in n of degree at most d − 1. Moreover, we show that the growth is bounded by a polynomial if and only if every element of the semigroup generated by A has all of its eigenvalues on or inside the unit circle.