  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 $T^*$ are subspace-hypercyclic operators. Finally, we give an $\mathcal
M$-hypercyclic criterion for an operator $T$ in terms of its eigenvalues.
