Subspace-supercyclic abelian linear semigroups

Let G be an abelian semigroup of matrices on $${\mathbb {K}}^{n}$$ K n ( $$n\ge 1$$ n ≥ 1 ), $${\mathbb {K}} = {\mathbb {R}} \text { or } {\mathbb {C}}$$ K = R or C . We say that G is subspace-supercyclic for a non-zero subspace M of $${\mathbb {K}}^{n}$$ K n , if there exists $$x\in {\mathbb {K}}^{n}$$ x ∈ K n such that $${\mathbb {K}}G(x)\cap M$$ K G ( x ) ∩ M is dense in M. G is called subspace-supercyclic if there exists a subspace M of $${\mathbb {K}}^{n}$$ K n such that G is subspace-supercyclic for M and dim $$(M)>1$$ ( M ) > 1 . We give a characterization of a subspace-supercyclic abelian semigroup of matrices on $${\mathbb {K}}^{n}$$ K n . For finitely generated semigroups, we give an effective spectral condition of checking that a given abelian semigroup is subspace-supercyclic. In particular, contrarily to matrices on $${\mathbb {R}}^{n}$$ R n , no matrix on $${\mathbb {C}}^{n}$$ C n , $$n\ge 2$$ n ≥ 2 , can be subspace-supercyclic. On the other hand, we show that if G of lenght $$\ge 2$$ ≥ 2 is supercyclic, then it is subspace-supercyclic. However, we provide for every $$n\ge 3$$ n ≥ 3 , a nontrivial example of abelian semigroup of matrices on $${\mathbb {K}}^{n}$$ K n of lenght 1, which is supercyclic but not subspace-supercyclic for any nontrivial subspace of $${\mathbb {K}}^{n}$$ K n . Further results on subspace-supercyclicity with prescribed subspaces are also given.