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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.

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

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