# Journal of Operator Theory

Volume 71, Issue 2, Spring 2014 pp. 479-490.

On multi-hypercyclic abelian semigroups of matrices on
$\mathbb{R}^{n}$

**Authors**:
Adlene Ayadi (1) and Habib Marzougui (2)

**Author institution:** (1) Department of Mathematics, Faculty of Science
of Gafsa, University of Gafsa, Gafsa, 2112, Tunisia

(2) Department of Mathematics, Faculty of Science of
Bizerte, University of Carthage, Jarzouna, 7021, Tunisia

**Summary: ** Let $G$ be an abelian semigroup of matrices on
$\mathbb{R}^{n}$ ($n\geqslant 1$). We show that $G$ is multi-hypercyclic if and only if it
has a somewhere dense orbit. We also give a necessary and sufficient condition for
a multi-hypercyclic semigroup $G$ to be hypercyclic, in terms of the index of
$G$ corresponding to negative eigenvalues of elements of $G$. On the other hand, we prove that
the closure $\overline{G(u)}$ of a somewhere dense orbit $G(u)$, $u\in
\mathbb{R}^{n}$, is invariant under multiplication by positive scalars; this answer a question
raised by Feldman. We also prove that $G^{k}$
is multi-hypercyclic for every $k\in \mathbb{N}^{p}$, ($p\in
\mathbb{N}$) whenever $G$ is multi-hypercyclic.

**DOI: **http://dx.doi.org/10.7900/jot.2012jun26.1981

**Keywords: ** hypercyclic, matrices, multi-hypercyclic, dense orbit,
semigroup, abelian

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