On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple

Details On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple

2009-04-21

arxiv.org/abs/0904.3142v2

Mathematics

Functional Analysis

15 pages, title changed, section for infinite dimensional spaces added

Scientific paper

In this paper we extend the notion of a locally hypercyclic operator to that of a locally hypercyclic tuple of operators. We then show that the class of hypercyclic tuples of operators forms a proper subclass to that of locally hypercyclic tuples of operators. What is rather remarkable is that in every finite dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$, a pair of commuting matrices exists which forms a locally hypercyclic, non-hypercyclic tuple. This comes in direct contrast to the case of hypercyclic tuples where the minimal number of matrices required for hypercyclicity is related to the dimension of the vector space. In this direction we prove that the minimal number of diagonal matrices required to form a hypercyclic tuple on $\mathbb{R}^n$ is $n+1$, thus complementing a recent result due to Feldman.

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