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4 - Weakly mixing operators

Published online by Cambridge University Press:  10 December 2009

Frédéric Bayart
Affiliation:
Université de Clermont-Ferrand II (Université Blaise Pascal), France
Étienne Matheron
Affiliation:
Université d'Artois, France
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Summary

Introduction

In Chapter 3, we saw that hypercyclicity is a rather “rigid” property: if T is hypercyclic then so is Tp for any positive integer p and so is ⋋T for any ⋋ ∈ T. In the same spirit, it is natural to ask whether TT remains hypercyclic. In topological dynamics, this property is quite well known.

DEFINITION Let X be a topological space. A continuous map T : XX is said to be (topologically) weakly mixing if T × T is topologically transitive on X × X.

Here, T×T : X×XX×X is the map defined by (T×T)(x, y) = (T(x), T(y)). When T is a linear operator, we identify T×T with the operator TT ∈ L(XX). We note that, by Birkhoff's transitivity theorem 1.2 and the remarks following it, one can replace “topologically transitive” by “hypercyclic” in the above definition if the underlying topological space X is a second-countable Baire space with no isolated points. In particular, a linear operator T on a separable F-space is weakly mixing iff TT is hypercyclic.

By definition, weakly mixing maps are topologically transitive. In the topological setting, it is easy to see that the converse is not true: for example, any irrational rotation of the circle T is topologically transitive but such a rotation is never weakly mixing. In the linear setting, things become very interesting because weak mixing turns out to be equivalent to the Hypercyclicity Criterion.

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Publisher: Cambridge University Press
Print publication year: 2009

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