A gap result for the norms of semigroups of matrices Original Research Article

Let short parallel·short parallel be a matrix norm on image and let image be a finite set of matrices in image. We define image to be the maximum norm of a product of n elements of image. We show that there is a gap in the possible growth of image, showing that image 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 image has all of its eigenvalues on or inside the unit circle.