Subspace-Hypercyclic Abelian Semigroups of Matrices on ${\mathbb {R}}^{n}$

Abstract

Let G be an abelian semigroup of matrices on ${\mathbb {K}}^{n}$ ($n\ge 1$), ${\mathbb {K}} = {\mathbb {R}} \text { or } {\mathbb {C}}$. It is called subspace-hypercyclic for a non-zero subspace M of ${\mathbb {K}}^{n}$, if there exists $x\in {\mathbb {K}}^{n}$ such that $G(x)\cap M$ is dense in M. In this paper, we give a complete characterization of subspace-hypercyclicity for abelian semigroups G of matrices on ${\mathbb {R}}^{n}$, $n\ge 1$ in terms of spectral density property. The complex case is done by the authors in Herzi and Marzougui (J Math Anal Appl 487:123960, 2020). In particular, we construct for any $n\ge 2$, a one parameter family of abelian semigroups which are subspace-hypercyclic for a plane, but not subspace-hypercyclic for any straight line in ${\mathbb {R}}^{n}$. Others examples of abelian semigroups of matrices on ${\mathbb {R}}^{n}$ satisfying some special properties are given. Further, we construct for every non-zero subspace M of ${\mathbb {K}}^{n}$, $n\ge 1$, a hypercyclic abelian semigroup of $\mathrm {GL}(n,{\mathbb {K}})$ that is subspace-hypercyclic for M. This result can be seen as the analogous for abelian semigroups to that given for operators on Banach spaces in Augusto and Pellegrini (Adv Oper Theory 5:1814–1824, 2020).