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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