Operators with diskcyclic vectors subspaces

Abstract

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 (Formula presented.). Also, if T is a diskcyclic operator and |λ| ≤ 1, then T − λI has dense range. Moreover, we prove that if α > 1, then (Formula presented.) is hypercyclic in a separable Hilbert space (Formula presented.) if and only if (Formula presented.) is diskcyclic in (Formula presented.). 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.