We study if the set of hypercyclic vectors of a hypercyclic operator is the complement of a σ-porous set. This leads to interesting results for both points of view: a limitation of the size of hypercyclic vectors, and new examples of first category sets which are not σ-porous.

In this paper, we give a brief review concerning diskcyclic operators and then we provide some further characterizations of diskcyclic operators on separable Hilbert spaces. In particular, we show that if x ∈ H has a disk orbit under T that is somewhere dense in H then the disk orbit of x under T need not be everywhere dense in H. We also show that the inverse and the adjoint of a diskcyclic operator need not be diskcyclic. Moreover, we establish another diskcyclicity criterion and use it to find a necessary and sufficient condition for unilateral backward shifts that are diskcyclic operators. We show that a diskcyclic operator exists on a Hilbert space H over the field of complex numbers if and only if dim(H) = 1 or dim(H) = ∞ . Finally we give a sufficient condition for the somewhere density disk orbit to be everywhere dense.

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

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