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 gave conditions for the unweighted unilateral backward shift on l2 to be hypercyclic. 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 G 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 ) 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 ) belongs to 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 the sets N(x;U) and N(U; V ) have 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 in X such that N(x;U) belongs to F. Let BD the collection of sets in N with positive upper Banach density. We generalize the main result of a recent paper of 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 N (the Stone-Cech 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 in X such that for any U non-empty open set in X, the set N(x;U) belongs to 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: a weighted backward shift Bw is F-operator if and only if (Bw; ... ;Bw^r) is d-F, for any r in N, where F runs along some filters strictly containing the family of cofinite 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; T^2) 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.
|Titolo:||RECURRENCE IN LINEAR DYNAMICS|
|Data di pubblicazione:||9-dic-2014|
|Parole Chiave:||hypercyclic operator; essential idempotent; reiteratively hypercyclic operator|
|Settore Scientifico Disciplinare:||Settore MAT/05 - Analisi Matematica|
|Citazione:||RECURRENCE IN LINEAR DYNAMICS ; supervisore non afferente all'ateneo: A. Peris. - Milano : Università degli studi di Milano. Università degli Studi di Milano, 2014 Dec 09. ((27. ciclo, Anno Accademico 2014.|
|Digital Object Identifier (DOI):||10.13130/puig-de-dios-yunied_phd2014-12-09|
|Appare nelle tipologie:||Tesi di dottorato|