Subspace hypercyclicity

TL;DR

This paper explores the concept of subspace-hypercyclicity in bounded linear operators on Hilbert spaces, providing examples, criteria, and spectral properties, and establishing its distinction from hypercyclicity and limitations for certain operator classes.

Contribution

It introduces the notion of subspace-hypercyclicity, constructs examples, and develops criteria and spectral insights, highlighting differences from classical hypercyclicity.

Findings

Subspace-hypercyclicity is a strictly infinite-dimensional phenomenon.

A Kitai-like criterion implies subspace-hypercyclicity.

Compact or hyponormal operators cannot be subspace-hypercyclic.

Abstract

A bounded linear operator T on Hilbert space is subspace-hypercyclic for a subspace M if there exists a vector whose orbit under T intersects the subspace in a relatively dense set. We construct examples to show that subspace-hypercyclicity is interesting, including a nontrivial subspace-hypercyclic operator that is not hypercyclic. There is a Kitai-like criterion that implies subspace-hypercyclicity and although the spectrum of a subspace-hypercyclic operator must intersect the unit circle, not every component of the spectrum will do so. We show that, like hypercyclicity, subspace-hypercyclicity is a strictly infinite-dimensional phenomenon. Additionally, compact or hyponormal operators can never be subspace-hypercyclic.