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 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.

Introduction

In this chapter, we prove several striking results showing that hypercyclicity is far from being an exotic phenomenon. In particular, we show that hypercyclic operators exist on any infinite-dimensional separable Fréchet space and that one can construct hypercyclic operators with “arbitrary” orbits. In the same spirit, we show that linear dynamics is in some sense as complicated as topological dynamics. We also discuss the size of the set of all hypercyclic operators on some given space X. Finally, we show that any Hilbert space operator may be written as the sum of two hypercyclic operators.

Mixing operators

When considering a strategy for constructing hypercyclic operators on an arbitrary “abstract” separable Banach space, one certainly has to keep in mind that some spaces have very few operators. Indeed, it is now well known that there exist (infinite dimensional) Banach spaces on which any operator has the form ⋋I + S, where S is strictly singular (for example, the so-called hereditarily indecomposable spaces; see Chapter 6 for some results concerning hypercyclic operators on such spaces). By a recent result of S. A. Argyros and R. Haydon [9], one can even encounter Banach spaces on which every operator has the form ⋋I + K, where K is a compact operator.

The examples of hypercyclic operators encountered in Chapter 1 are very far from being small perturbations of ⋋I. Yet, if hypercyclic operators are supposed to exist on any separable Banach space, it must be true that some operators of the form ⋋I + K are hypercyclic.

Introduction

In this final chapter, our aim is to give a short and gentle introduction to the kind of operator constructed by C. J. Read in the 1980s.

In his 1987 paper [202], Read solved in the negative the invariant subset problem for the space l1(N), and in fact for any separable Banach space containing a complemented copy of l1. In other words, he was able to produce on such a space an operator for which every non-zero vector is hypercyclic.

The construction carried out in [202] is something of a tour de force. Moreover, its understanding requires some familiarity with earlier constructions by the same author relating to the invariant subspace problem, which are already quite involved (see e.g. [201]). This convinced us that we should not be overly ambitious regarding the material presented in this chapter. Thus, we have chosen to concentrate on the simplest example of a “Read-type” operator, i.e. an operator T acting on l1(N) for which every non-zero vector x is cyclic. This is a counter-example to the invariant subspace problem for the space l1.

As already said, we have tried to give a helpful presentation, meaning that we have made some effort to explain the underlying ideas as we understand them, inserting heuristic comments whenever this seemed necessary. Some parts of the discussion are deliberately informal, but the construction is nevertheless complete and selfcontained. We hope that this chapter will be useful for people interested in that kind of question.

Introduction

In Chapter 1, we observed that the set of hypercyclic vectors of any hypercyclic operator T has a rich algebraic structure: there exists a dense linear subspace of the underlying space X consisting entirely of hypercyclic vectors, except 0.

It may also happen that HC(T) contains a closed infinite-dimensional subspace (except 0). This is the topic that we will consider in this chapter. For brevity, we adopt the following terminology: by a hypercyclic subspace for an operator T ∈ L(X), we shall always mean a closed infinite-dimensional subspace Z ⊂ X such that Z \ {0} ⊂ HC(T).

Our main goal in this chapter is to prove the two theorems stated below, which show that hypercyclic subspaces exist under quite natural assumptions. For simplicity, we will restrict ourselves to the case of Banach space operators.

The following basic result is due to A. Montes-Rodríguez [183].

THEOREM 8.1 Let X be a separable Banach space, and let T ∈ L(X). Assume that the following hold for some increasing sequence of integers (nk).

1. T satisfies the Hypercyclicity Criterion with respect to (nk).

2. There exists a closed infinite-dimensional subspace E ⊂ X such that Tnk (x) → 0 for all x ∈ E.

Then T has a hypercyclic subspace.

It is important to note that properties (1) and (2) in Theorem 8.1 have to be satisfied by the same sequence (nk). However, it turns out that this restriction is in fact not necessary.

Linear dynamics is a young and rapidly evolving branch of functional analysis, which was probably born in 1982 with the Toronto Ph.D. thesis of C. Kitai [158]. It has become rather popular, thanks to the efforts of many mathematicians. In particular, the seminal paper [123] by G. Godefroy and J. H. Shapiro, the authoritative survey [133] by K.-G. Grosse-Erdmann and the beautiful notes [222] by J. H. Shapiro have had a considerable influence on both its internal development and its diffusion within the mathematical community. After more than two decades of active research, this would seem to be the proper time to write a book about it.

As the name indicates, linear dynamics is mainly concerned with the behaviour of iterates of linear transformations. On finite-dimensional spaces, things are rather well understood since linear transformations are completely described by their Jordan canonical form. However, a new phenomenon appears in an infinite-dimensional setting: linear operators may have dense orbits. In fact, quite a lot of natural operators have this property.

To settle some terminology, let us recall that if T is a continuous linear operator acting on some topological vector space X, the T-orbit of a vector x ∈ X is the set O(x, T) := {x, T(x), T2(x), … }. The operator T is said to be hypercyclic if there exists some vector x ∈ X whose T-orbit is dense in X. Such a vector x is said to be hypercyclic for T.