Operators with diskcyclic vectors subspaces

In this paper, we prove that if T is a diskcyclic operator then the closed unit disk multiplied by the union of the numerical range of all iterations of T   is dense in HH. Also, if T is a diskcyclic operator and |λ| ≤ 1, then T − λI has dense range. Moreover, we prove that if α > 1, then 1αT is hypercyclic in a separable Hilbert space HH if and only if T⊕αIℂ is diskcyclic in H⊕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.