Some properties of subspaces-hypercyclic operators

Abstract

In this paper, we answer a question posed in the introduction of \cite{sub hyp} positively, i.e, we show that if $T$ is $\mathcal{M}$-hypercyclic operator with $\mathcal{M}$-hypercyclic vector $x$ in a Hilbert space $\mathcal{H}$, then $P(Orb(T,x))$ is dense in the subspace $\mathcal{M}$ where $P$ is the orthogonal projection onto $\mathcal{M}$. Furthermore, we give some relations between ${\mathcal{M}}^{\perp }$-hypercyclicity and the orthogonal projection onto ${\mathcal{M}}^{\perp }$. We also give sufficient conditions for a bilateral weighted shift operators on a Hilbert space ${\ell }^2(\mathbb{Z})$ to be subspace-hypercyclic, cosequently, there exists an operator $T$ such that both $T$ and $T^{\ast }$ are subspace-hypercyclic operators. Finally, we give an $\mathcal{M}$-hypercyclic criterion for an operator $T$ in terms of its eigenvalues.