  In this paper, we prove that if $T$ is diskcyclic operator then the closed
unit disk multiplied by the union of the numerical range of all iterations of
$T$ is dense in $\mathcal H$. Also, if $T$ is diskcyclic operator and
$|\lambda|\le 1$, then $T-\lambda I$ has dense range. Moreover, we prove that
if $\alpha >1$, then $\frac{1}{\alpha}T$ is hypercyclic in a separable Hilbert
space $\mathcal H$ if and only if $T \oplus \alpha I_{\mathbb{C}}$ is
diskcyclic in $\mathcal H \oplus \mathbb{C}$. We show at least in some cases a
diskcyclic operator has an invariant, dense linear subspace or an infinite
dimensional closed linear subspace, whose non-zero elements are diskcyclic
vectors. However, we give some counterexamples to show that not always a
diskcyclic operator has such a subspace.
