A note on a gap result for norms of semigroups of matrices

Let ∥·∥ be a matrix norm and Σ be a bounded set of n × n complex matrices. For each m = 1, 2, …, let , where Σm = {A1A2⋯Am: Ai ∈ Σ}. Bell proved that there is a gap in the possible growth of if Σ consists of finitely many n × n complex matrices. In this note, based on a lemma by Elsner, we give an elementary proof of Bell’s theorem in bounded case.