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Operators with Diskcyclic Vectors Subspaces

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 $|λ|\le 1$, then $T-λI$ has dense range. Moreover, we prove that if $α>1$, then $\frac{1}αT$ is hypercyclic in a separable Hilbert space $\mathcal H$ if and only if $T \oplus αI_{\mathbb{C}}$ 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.