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Master's Dissertation
DOI
10.11606/D.45.2015.tde-01102015-120053
Document
Author
Full name
Andre Quintal Augusto
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2015
Supervisor
Committee
Rodrigues, Leonardo Pellegrini (President)
Fajardo, Rogerio Augusto dos Santos
Kaufmann, Pedro Levit
Title in Portuguese
Operadores hipercíclicos e o critério de hiperciclicidade
Keywords in Portuguese
Critério de hiperciclicidade
Hiperciclicidade
Operadores hipercíclicos.
Abstract in Portuguese
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.
Title in English
Hypercyclic operators and the hypercyclicity criterion
Keywords in English
Hypercyclic operators
Hypercyclicity
Hypercyclicity criterion
Abstract in English
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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Publishing Date
2015-10-08
 
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