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. / Puig De Dios, Y. (2014). Recurrence in Linear Dynamics [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/48473

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In this dissertation, we explore the nature of Schur rings over finite cyclic groups, both algebraically and combinatorially. We provide a survey of many fundamental properties and constructions of Schur rings over arbitrary finite groups. After specializing to the case of cyclic groups, we provide an extensive treatment of the idempotents of Schur rings and a description for the complete set of primitive idempotents. We also use Galois theory to provide a classification theorem of Schur rings over cyclic groups similar to a theorem of Leung and Man and use this classification to provide a formula for the number of Schur rings over cyclic p-groups.

This thesis is chiefly concerned with a classical conjecture of Littlewood's regarding the L¹-norm of the Fourier transform, and the closely related idem-potent theorem. The vast majority of the results regarding these problems are, in some sense, qualitative or at the very least infinitary and it has become increasingly apparent that a quantitative state of affairs is desirable. Broadly speaking, the first part of the thesis develops three new tools for tackling the problems above: We prove a new structural theorem for the spectrum of functions in A(G); we extend the notion of local Fourier analysis, pioneered by Bourgain, to a much more general structure, and localize Chang's classic structure theorem as well as our own spectral structure theorem; and we refine some aspects of Freiman's celebrated theorem regarding the structure of sets with small doubling. These tools lead to improvements in a number of existing additive results which we indicate, but for us the main purpose is in application to the analytic problems mentioned above. The second part of the thesis discusses a natural version of Littlewood's problem for finite abelian groups. Here the situation varies wildly with the underlying group and we pay special attention first to the finite field case (where we use Chang's Theorem) and then to the case of residues modulo a prime where we require our new local structure theorem for A(G). We complete the consideration of Littlewood's problem for finite abelian groups by using the local version of Chang's Theorem we have developed. Finally we deploy the Freiman tools along with the extended Fourier analytic techniques to yield a fully quantitative version of the idempotent theorem.

Cette thèse étudie des problèmes liés aux treillis des sous-groupes quantiques et la caractérisationdes actions ergodiques et des états idempotents d’un groupe quantique compact.Elle consiste en 3 parties. La première partie présente des résultats préliminaires sur lesgroupes quantiques localement compacts, les sous-groupes quantiques normaux ainsi queles actions ergodiques et les états idempotents. La seconde partie étudie l’analogue quantiquede la règle de modularité de Dedekind et de l’analogue quantique des théorèmesd’isomorphisme de Noether ainsi que leur conséquences comme le théorème de raffinementde Schreier, et le théorème Jordan-Hölder. Cette partie s’inspire du travail de recherche deShuzhouWang sur l’analogue quantique du troisième théorème d’isomorphisme de Noetherpour les groupes quantiques compacts ainsi que le travail récent de Kasprzak, Khosraviet Soltan sur l’analogue quantique du premier théorème d’isomorphisme de Noether pourles groupes quantiques localement compacts. Dans la troisième partie, nous caractérisonsles états idempotents du groupe quantique compact O−1(2) en s’appuyant sur la caractérisationde ses actions ergodiques plongeables. Cette troisième partie est dans la lignedes travaux fait par Franz, Skalski et Tomatsu pour les groupes quantiques compactsUq(2), SUq(2) et SOq(3). Nous classifions au préalable les actions ergodiques et les actionsergodiques plongeables du groupe quantique compact O−1(2).Les travaux présentés dans cette thèse se basent sur deux articles de l’auteur et al.Le premier s’intitule “Fundamental isomorphism theorems for quantum groups” et a étéaccepté pour publication dans Expositionae Mathematicae et le second est intitulé “Ergodicactions and idempotent states of O−1(2)” et est en cours de finalisation pour être soumis. / This thesis studies problems linked to the lattice of quantum subgroups and characterizationof ergodic actions and idempotent states of a compact quantum group. It consistsof three parts. The first part present some preliminary results about locally compactquantum groups, normal quantum subgroups, ergodic actions and idempotent states. Thesecond part studies the quantum analog of Dedekind’s modularity law, Noether’s isomorphismtheorem and their consequences as the Schreier refinement theorem and theJordan-Hölder theorem. This part completes the work of Shuzhou WANG on the quantumanalog of the third isomorphism theorem for compact quantum group and the recentwork of Kasprzak, Khosravi and Soltan on the quantum analog of the first Noether isomorphismtheorem for locally compact quantum groups. In the third part, we characterizeidempotent states of the compact quantum group O−1(2) relying on the characterizationof embeddable ergodic actions. This third part is in the sequence of the seminal works ofFranz, Skalski and Tomatsu for the compact quantum groups Uq(2), SUq(2) and SOq(3).We classify in advance the ergodic actions and embeddable ergodic actions of the compactquantum group O−1(2).This thesis is based on two papers of the author and al. The first one is entitled“Fundamental isomorphism theorems for quantum groups” which have been accepted forpublication in Expositionae Mathematicae and the second one is entitled “Ergodic actionsand idempotent states of O−1(2)” and is being finalized for submission.

Chapter 1 reviews better-known factorization theorems of a square matrix. For example, a square matrix over a field can be expressed as a product of two symmetric matrices; thus square matrices over real numbers can be factorized into two diagonalizable matrices. Factorizing matrices over complex num hers into Hermitian matrices is discussed. The chapter concludes with theorems that enable one to prescribe the eigenvalues of the factors of a square matrix, with some degree of freedom. Chapter 2 proves that a square matrix over arbitrary fields (with one exception) can be expressed as a product of two diagona lizab le matrices. The next two chapters consider decomposition of singular matrices into Idempotent matrices, and of nonsingutar matrices into Involutions. Chapter 5 studies factorization of a comp 1 ex matrix into Positive-( semi )definite matrices, emphasizing the least number of such factors required / Mathematical Sciences / M.Sc. (MATHEMATICS)

The axiomatic theory of ` Zelazko defines a variety of general spectra where specified axioms are satisfied. However, there arise a number of spectra, usually defined for a single element of a Banach algebra, that are not covered by the axiomatic theory of ` Zelazko. V. Kordula and V. M¨uller addressed this issue and created the theory of regularities. Their unique idea was to describe the underlying set of elements on which the spectrum is defined. The axioms of a regularity provide important consequences. We prove that the set of Koliha-Drazin invertible elements, which includes the Drazin invertible elements, forms a regularity. The properties of the spectrum corresponding to a regularity are also investigated. / Mathematical Sciences / M. Sc. (Mathematics)

Chapter 1 reviews better-known factorization theorems of a square matrix. For example, a square matrix over a field can be expressed as a product of two symmetric matrices; thus square matrices over real numbers can be factorized into two diagonalizable matrices. Factorizing matrices over complex numbers into Hermitian matrices is discussed. The chapter concludes with theorems that enable one to prescribe the eigenvalues of the factors of a square matrix, with some degree of freedom. Chapter 2 proves that a square matrix over arbitrary fields (with one exception) can be expressed as a product of two diagonalizable matrices. The next two chapters consider decomposition of singular matrices into Idempotent matrices, and of nonsingular matrices into Involutions. Chapter 5 studies factorization of a complex matrix into Positive-(semi)definite matrices, emphasizing the least number of such factors required. / Mathematical Sciences / M. Sc. (Mathematics)

Chapter 1 reviews better-known factorization theorems of a square matrix. For example, a square matrix over a field can be expressed as a product of two symmetric matrices; thus square matrices over real numbers can be factorized into two diagonalizable matrices. Factorizing matrices over complex num hers into Hermitian matrices is discussed. The chapter concludes with theorems that enable one to prescribe the eigenvalues of the factors of a square matrix, with some degree of freedom. Chapter 2 proves that a square matrix over arbitrary fields (with one exception) can be expressed as a product of two diagona lizab le matrices. The next two chapters consider decomposition of singular matrices into Idempotent matrices, and of nonsingutar matrices into Involutions. Chapter 5 studies factorization of a comp 1 ex matrix into Positive-( semi )definite matrices, emphasizing the least number of such factors required / Mathematical Sciences / M.Sc. (MATHEMATICS)

The axiomatic theory of ` Zelazko defines a variety of general spectra where specified axioms are satisfied. However, there arise a number of spectra, usually defined for a single element of a Banach algebra, that are not covered by the axiomatic theory of ` Zelazko. V. Kordula and V. M¨uller addressed this issue and created the theory of regularities. Their unique idea was to describe the underlying set of elements on which the spectrum is defined. The axioms of a regularity provide important consequences. We prove that the set of Koliha-Drazin invertible elements, which includes the Drazin invertible elements, forms a regularity. The properties of the spectrum corresponding to a regularity are also investigated. / Mathematical Sciences / M. Sc. (Mathematics)