On Read's type operators on Hilbert spaces

TL;DR

This paper constructs specific operators on Hilbert spaces using Read's method, demonstrating diverse behaviors of hypercyclic vectors and orbit structures, including examples with large hypercyclic sets, orbit-unicellularity, and non-orbit-reflexivity.

Contribution

It introduces novel Hilbert space operators with unique hypercyclic and orbit properties, expanding understanding of operator dynamics.

Findings

Existence of operators with closed orbit closures

Operators with non-hypercyclic vectors forming a Gauss null set

Examples of orbit-unicellular and non-orbit-reflexive operators

Abstract

Using Read's construction of operators without non-trivial invariant subspaces/subsets on $\ell_{1}$ or $c_{0}$, we construct examples of operators on a Hilbert space whose set of hypercyclic vectors is "large" in various senses. We give an example of an operator such that the closure of every orbit is a closed subspace, and then, answering a question of D. Preiss, an example of an operator such that the set of its non-hypercyclic vectors is Gauss null. This operator has the property that it is orbit-unicellular, i.e. the family of the closures of its orbits is totally ordered. We also exhibit an example of an operator on a Hilbert space which is not orbit-reflexive.