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On Orbits of Operators on Hilbert SpaceSmith, Lidia 2009 August 1900 (has links)
In this dissertation we treat some problems about possible density of orbits for
non-hypercyclic operators and we enlarge the class of known non-orbit-transitive operators.
One of the questions related to hypercyclic operators that we answer is
whether the density (in the set of positive real numbers) of the norms of the elements
in the orbit for each nonzero vector in the Hilbert space is sufficient to imply that
at least one vector has orbit dense in the Hilbert space. We show that the density
of the norms is not a sufficient condition to imply hypercyclicity by constructing a
weighted bilateral shift that, on one hand, satisfies the orbit-density property (in the
sense defined above), but, on the other hand, fails to be hypercyclic. The second
major topic that we study refers to classes of operators that are not hypertransitive
(or orbit-transitive) and is related to the invariant subspace problem on Hilbert space.
It was shown by Jung, Ko and Pearcy in 2005 that every compact perturbation of
a normal operator is not hypertransitive. We extend this result, after introducing
the related notion of weak hypertransitivity, by giving a sufficient condition for an
operator to belong to the class of non-weakly-hypertransitive operators. Next, we
study certain 2-normal operators and their compact perturbations. In particular, we
consider operators with a slow growth rate for the essential norms of their powers.
Using a new idea, of accumulation of growth for each given power on a set of different
orthonormal vectors, we establish that the studied operators are not hypertransitive.
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Dense Orbits of the Aluthge TransformRion, Kevin 25 March 2011 (has links)
No description available.
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Hypercyclic Algebras and Affine DynamicsPapathanasiou, Dimitrios 10 April 2017 (has links)
No description available.
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Hypercyclic Extensions of an Operator on a Hilbert Subspace with Prescribed BehaviorsKadel, Gokul Raj 26 July 2013 (has links)
No description available.
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Recurrence in Linear DynamicsPuig de Dios, Yunied 30 March 2015 (has links)
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 no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/48473 / TESIS
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A Strictly Weakly Hypercyclic Operator with a Hypercyclic SubspaceMadarasz, Zeno 11 August 2023 (has links)
No description available.
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Disjoint Hypercyclic and Supercyclic Composition OperatorsMartin, Ozgur 04 August 2010 (has links)
No description available.
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Hypercyclic Extensions Of Bounded Linear OperatorsTurcu, George R. 20 December 2013 (has links)
No description available.
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Operadores hipercíclicos e o critério de hiperciclicidade / Hypercyclic operators and the hypercyclicity criterionAugusto, Andre Quintal 03 August 2015 (has links)
Dado um espaço vetorial topológico $X$ e um operador linear $T$ contínuo em $X$, dizemos que $T$ é {\\it hipercíclico} se, para algum $y \\in X$, o conjunto $\\{y, T(y), T^2(y), T^3(y), \\ldots T^n(y) \\ldots \\}$ for denso em $X$. Um dos principais resultados envolvendo operadores hipercíclicos consiste no chamado {\\it Critério de Hiperciclicidade}. Tal Critério fornece uma condição suficiente para que um operador linear contínuo seja hipercíclico. Por muitos anos, procurou-se saber se o Critério também era uma condição necessária. Em \\cite, Bayart e Matheron construíram, nos espaços de Banach clássicos $c_0$ e $\\ell_p, 1 \\leq p < \\infty$, um operador hipercíclico $T$ que não satisfaz o Critério. Neste trabalho, apresentamos a construção realizada por Bayart e Matheron. Além disso, também apresentamos alguns resultados sobre hiperciclicidade. / Given a topological vector space $X$ and a continuous linear operator $T$, we say that $T$ is {\\it hypercylic} if, for some $y \\in X$, the set $\\{y, T(y), T^2(y), T^3(y), \\ldots T^n(y) \\ldots \\}$ is dense in $X$. One of the main results concerning hypercyclic operators is the so-called {\\it Hypercyclicity Criterion}. Such Criterion gives a sufficient condition to a continuous linear operator be hypercyclic. For many years, it sought to know if the Criterion was also a necessary condition. In \\cite, Bayart and Matheron constructed, in the classical Banach spaces $c_0$ e $\\ell_p, 1 \\leq p < \\infty$, a hypercyclic operator $T$ which doesn\'t satisfy the Criterion. In this work, we present the Bayart/Matheron construction. We also present some results about hypercyclicity.
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Operadores hipercíclicos e o critério de hiperciclicidade / Hypercyclic operators and the hypercyclicity criterionAndre Quintal Augusto 03 August 2015 (has links)
Dado um espaço vetorial topológico $X$ e um operador linear $T$ contínuo em $X$, dizemos que $T$ é {\\it hipercíclico} se, para algum $y \\in X$, o conjunto $\\{y, T(y), T^2(y), T^3(y), \\ldots T^n(y) \\ldots \\}$ for denso em $X$. Um dos principais resultados envolvendo operadores hipercíclicos consiste no chamado {\\it Critério de Hiperciclicidade}. Tal Critério fornece uma condição suficiente para que um operador linear contínuo seja hipercíclico. Por muitos anos, procurou-se saber se o Critério também era uma condição necessária. Em \\cite, Bayart e Matheron construíram, nos espaços de Banach clássicos $c_0$ e $\\ell_p, 1 \\leq p < \\infty$, um operador hipercíclico $T$ que não satisfaz o Critério. Neste trabalho, apresentamos a construção realizada por Bayart e Matheron. Além disso, também apresentamos alguns resultados sobre hiperciclicidade. / Given a topological vector space $X$ and a continuous linear operator $T$, we say that $T$ is {\\it hypercylic} if, for some $y \\in X$, the set $\\{y, T(y), T^2(y), T^3(y), \\ldots T^n(y) \\ldots \\}$ is dense in $X$. One of the main results concerning hypercyclic operators is the so-called {\\it Hypercyclicity Criterion}. Such Criterion gives a sufficient condition to a continuous linear operator be hypercyclic. For many years, it sought to know if the Criterion was also a necessary condition. In \\cite, Bayart and Matheron constructed, in the classical Banach spaces $c_0$ e $\\ell_p, 1 \\leq p < \\infty$, a hypercyclic operator $T$ which doesn\'t satisfy the Criterion. In this work, we present the Bayart/Matheron construction. We also present some results about hypercyclicity.
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