Weakly mixing operators

Abstract

Introduction In Chapter 3, we saw that hypercyclicity is a rather “rigid” property: if T is hypercyclic then so is T p for any positive integer p and so is ⋋ T for any ⋋ ∈ T. In the same spirit, it is natural to ask whether T ⊕ T remains hypercyclic. In topological dynamics, this property is quite well known. DEFINITION Let X be a topological space. A continuous map T : X → X is said to be ( topologically ) weakly mixing if T × T is topologically transitive on X × X . Here, T × T : X × X → X × 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 T ⊕ T ∈ L( X ⊕ X ). 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 T ⊕ T 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.