Subspace-hypercyclic abelian linear semigroups

Abstract

We consider the subspace-hypercyclicity for abelian semigroups G of matrices on Kn (n≥1), K=RorC. We say that G is subspace-hypercyclic for a non-zero subspace M of Kn if there exists x∈Kn such that G(x)∩M is dense in M. We provide an effective method for checking that a given abelian semigroup is subspace-hypercyclic. We construct a nontrivial example of an abelian semigroup that has a dense orbit relative to a certain straight line but it is not subspace-hypercyclic for this straight line. For every n≥2, we also construct an example of an abelian semigroup generated by two matrices where the subspace is hypercyclic but not hypercyclic. We determine the minimal number of diagonalizable matrices over C that generate a subspace-hypercyclic abelian semigroup on Cn. Moreover, we prove that Gk is subspace-hypercyclic for every k∈N0p whenever G is a subspace-hypercyclic abelian semigroup of matrices on Cn.