A Review of Some Works in the Theory of Diskcyclic Operators

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 $$x\in {\mathcal {H}}$$ has a disk orbit under $$T$$ that is somewhere dense in $${\mathcal {H}}$$ , then the disk orbit of $$x$$ under $$T$$ need not be everywhere dense in $${\mathcal {H}}$$ . 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 $${\mathcal {H}}$$ over the field of complex numbers if and only if $$\dim ({\mathcal {H}})=1$$ or $$\dim ({\mathcal {H}})=\infty $$ . Finally, we give a sufficient condition for the somewhere density disk orbit to be everywhere dense.