Abelian Semigroups of Matrices on ${\mathbb {K}}^{n}$ and Convex-Cyclicity

Abstract

We extend the notion of convex-cyclicity for matrices to that of convex-cyclicity for abelian semigroups of matrices on ${\mathbb {K}}^{n}$. We say that an abelian semigroup G of matrices on ${\mathbb {K}}^{n}$ is convex-cyclic if there exists $x\in {\mathbb {K}}^{n}$ such that the convex hull of the orbit G(x) of x under G is dense in ${\mathbb {K}}^{n}$. We provide an effective method for checking that a given abelian semigroup is convex-cyclic. In particular we give a spectral characterization of convex-cyclicity for finitely generated abelian semigroups of diagonalizable matrices. We also obtain an example of a convex-cyclic abelian semigroup which does not contain a convex-cyclic matrix. We prove that there exists an abelian semigroup of matrices on ${\mathbb {K}}^{n}$ which is somewhere convex-cyclic but not convex-cyclic. This solves two questions of Rezaei in (Linear Algebra Appl 438:4190–4203, 2013) and Feldman and McGuire in (Oper Matrices 12(2):465–492, 2017) respectively. On the other hand, we provide the link between $\varepsilon$-hypercyclicity and convex-cyclicity for abelian semigroups of matrices on ${\mathbb {K}}^{n}$ for every $0<\varepsilon <1$.