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Resumen:
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A bounded and linear operator is said to be hypercyclic if there exists a
vector such that its orbit under the action of the operator is dense. The first
example of a hypercyclic operator on a Banach space was given in ...[+]
A bounded and linear operator is said to be hypercyclic if there exists a
vector such that its orbit under the action of the operator is dense. The first
example of a hypercyclic operator on a Banach space was given in 1969 by
Rolewicz who showed that if B is the unweighted unilateral backward shift
on l
2
, then λB is hypercyclic if and only if |λ| > 1. Among its features,
we can mention for example that finite-dimensional spaces cannot support
hypercyclic operators, proved by Kitai. On the other hand, several people
have shown in different contexts, in the Hilbert space frame, that the set of
hypercyclic vectors for a hypercyclic operator is a Gδ dense set.
This thesis is divided into four chapters. In the first one, we give some
preliminaries by mentioning some definitions and known results that will be
of great help later.
In chapter 2, we introduce a refinement of the notion of hypercyclicity,
relative to the set N(U, V ) = {n ∈ N : T
−nU ∩ V 6= ∅} when belonging
to a certain collection F of subsets of N, namely a bounded and linear
operator T is called F-operator if N(U, V ) ∈ F, for any pair of non-empty
open sets U, V in X. First, we do an analysis of the hierarchy established
between F-operators, whenever F covers those families mostly studied in
Ramsey theory. Second, we investigate which kind of properties of density
can have the sets N(x, U) = {n ∈ N : T
nx ∈ U} and N(U, V ) for a given
hypercyclic operator, and classify the hypercyclic operators accordingly to
these properties.
In chapter three, we introduce the following notion: an operator T on
X satisfies property PF if for any U non-empty open set in X, there exists
x ∈ X such that N(x, U) ∈ F. Let BD the collection of sets in N with positive
upper Banach density. We generalize the main result of a paper due to
Costakis and Parissis using a strong result of Bergelson and Mccutcheon in
the vein of Szemerédi’s theorem, leading us to a characterization of those operators
satisfying property PBD. It turns out that operators having property
PBD satisfy a kind of recurrence described in terms of essential idempotents
of βN (the Stone-Čech compactification of N). We will discuss the case of
weighted backward shifts satisfying property PBD. On the other hand, as
a consequence we obtain a characterization of reiteratively hypercyclic operators,
i.e. operators for which there exists x ∈ X such that for any U
non-empty open set in X, the set N(x, U) ∈ BD.
The fourth chapter focuses on a refinement of the notion of disjoint hypercyclicity.
We extend a result of Bès, Martin, Peris and Shkarin by stating:
Bw is F-weighted backward shift if and only if (Bw, . . . , Br
w) is d-F, for any
r ∈ N, where F runs along some filters containing strictly the family of cofi-
nite sets, which are frequently used in Ramsey theory. On the other hand,
we point out that this phenomenon does not occur beyond the weighted shift
frame by showing a mixing linear operator T on a Hilbert space such that the
tuple (T, T2
) is not d-syndetic. We also, investigate the relationship between
reiteratively hypercyclic operators and d-F tuples, for filters F contained
in the family of syndetic sets. Finally, we examine conditions to impose in
order to get reiterative hypercyclicity from syndeticity in the weighted shift
frame.
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