When is a scaled contraction hypercyclic?

Abstract

Hypercyclic operators are operators with dense orbits. A contraction cannot be hypercyclic since its orbits are bounded sets. Nevertheless, by multiplying a contraction with a scalar of absolute value larger than 1, the resulting scaled contraction can occasionally be a hypercyclic operator. In this paper, we investigate which Hilbert space contractions have that property and which don’t. We introduce the set $$\Lambda (T)$$ of all scalars which produce a hypercyclic operator, by scaling the operator T, and determine $$\Lambda (T)$$ in various cases. New properties of hyperciclic operators are discovered in this process. For instance, it is proved that any connected component of the essential spectrum of a hypercyclic operator must meet the unit circle.