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.