Hypercyclic tuples of operators on ℂⁿ and ℝⁿ

Abstract

A tuple (T ₁, … , T _n ) of commuting continuous linear operators on a topological vector space X is called hypercyclic if there is x ∈ X such that the orbit of x under the action of the semigroup generated by T ₁, … , T _n is dense in X. This concept was introduced by Feldman, who has raised seven questions on hypercyclic tuples. We answer four of those, which can be dealt with on the level of operators on finite dimensional spaces. In particular, we prove that the minimal cardinality of a hypercyclic tuple of operators on ℂ ⁿ (respectively, on ℝ ⁿ ) is n + 1 (respectively, ), that there are non-diagonalizable tuples of operators on ℝ² which possess an orbit being neither dense nor nowhere dense and construct a hypercyclic 6-tuple of operators on ℂ³ such that every operator commuting with each member of the tuple is non-cyclic.

Keywords:

• cyclic operators
• hypercyclic operators
• supercyclic operators
• universal families

AMS Subject Classifications::

• 47A16
• 37A25

Acknowledgements

The author is grateful to the referee for numerous helpful suggestions.