Mixing operators on spaces with weak topology

TL;DR

This paper characterizes mixing operators on spaces with weak topology by linking their mixing property to the absence of finite-dimensional invariant subspaces in their dual operators, extending known results on hypercyclicity.

Contribution

It provides a new characterization of mixing operators on weak topology spaces, connecting the property to the dual operator structure, and generalizes previous hypercyclicity results.

Findings

A continuous linear operator on a weak topology space is mixing iff its dual has no finite-dimensional invariant subspaces.

Characterization of hypercyclic operators on the space ω.

Demonstrates that for hypercyclic operators T on ω, T⊕T is also hypercyclic.

Abstract

We prove that a continuous linear operator $T$ on a topological vector space $X$ with weak topology is mixing if and only if the dual operator $T'$ has no finite dimensional invariant subspaces. This result implies the characterization of hypercyclic operators on the space $\omega$ due to Herzog and Lemmert and implies the result of Bayart and Matheron, who proved that for any hypercyclic operator $T$ on $\omega$, $T\oplus T$ is also hypercyclic.