A review of some works in the theory of diskcyclic operators

Abstract: 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.