A review of some works in the theory of diskcyclic operators

TL;DR

This paper reviews the theory of diskcyclic operators, providing new characterizations, criteria, and conditions for their existence and properties on Hilbert spaces, highlighting differences between dense and somewhere dense disk orbits.

Contribution

It introduces a new diskcyclicity criterion, characterizes diskcyclic operators on Hilbert spaces, and explores properties of their orbits and inverse operators.

Findings

Diskcyclic operators exist only on 1D or infinite-dimensional Hilbert spaces.

Somewhere dense disk orbits need not be everywhere dense.

Inverse and adjoint of diskcyclic operators may not be diskcyclic.

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.