Operators with Diskcyclic Vectors Subspaces

TL;DR

This paper investigates properties of diskcyclic operators, establishing conditions for their dense range, hypercyclicity, and the existence of invariant subspaces, along with counterexamples highlighting limitations of these properties.

Contribution

It provides new results on the structure and properties of diskcyclic operators, including their relation to hypercyclicity and invariant subspaces, with counterexamples illustrating limitations.

Findings

Diskcyclic operators have dense numerical range-related sets.

For || b1 1, T- I has dense range.

Scaling T by 1/ relates to hypercyclicity and diskcyclicity.

Abstract

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 $\mathcal H$. Also, if $T$ is diskcyclic operator and $|\lambda|\le 1$, then $T-\lambda I$ has dense range. Moreover, we prove that if $\alpha >1$, then $\frac{1}{\alpha}T$ is hypercyclic in a separable Hilbert space $\mathcal H$ if and only if $T \oplus \alpha I_{\mathbb{C}}$ is diskcyclic in $\mathcal H \oplus \mathbb{C}$. 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.