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Doctoral Thesis
Full name
Vinícius Colferai Corrêa Miranda
Knowledge Area
Date of Defense
São Paulo, 2022
Lourenço, Mary Lilian (President)
Albuquerque, Nacib André Gurgel e
Botelho, Geraldo Márcio de Azevedo
Dimant, Verónica Isabel
Fávaro, Vinícius Vieira
Title in Portuguese
Estudo de propriedades geométricas em reticulados de Banach
Keywords in Portuguese
Conjuntos almost Grothendieck
Conjuntos limited
Operadores positivos
Propriedade (d)
Reticulados de Banach
Abstract in Portuguese
Nesta tese trabalhamos no ambiente de espaços de Banach e de reticulados de Banach. Em um primeiro momento, estudamos a propriedade (d) em reticulados de Banach e a classe dos operadores alcc. Em particular, apresentamos uma caracterização de tal propriedade considerando a envoltória sólida de conjuntos almost limited e estudamos o problema do operador dominado para a classe dos operadores alcc. Em seguida, introduzimos duas novas classes de subconjuntos em reticulados de Banach. Tais classes são associadas às propriedades do tipo Grothendieck em reticulados. Comparamos essas classes de conjuntos com outras classes de conjuntos já conhecidos. Introduzimos também uma classe de operadores associados e obtivemos diversos resultados. É importante obter exemplos de reticulados de Banach que satisfaçam as propriedades geométricas já conhecidas nessa classe. Estudamos ainda tais propriedades nos reticulados (⨁_{n=1}^∞ \ell_2^n ight )_0, (⨁_{n=1}^∞ \ell_2^n ight )_1 e (⨁_{n=1}^∞ \ell_2^n ight )_∞. Por fim, voltando a estrutura de espaços de Banach, considerando sequências fraco-estrela $p$-somáveis no lugar de sequências fraco-estrela nulas, introduzimos e estudamos uma classe de conjuntos que é maior que a classe dos conjuntos limited. Também obtivemos resultados envolvendo classes de operadores associados.
Title in English
The study of geometric properties in Banach lattices
Keywords in English
Almost Grothendieck sets
Banach lattices
Limited sets
Positive operators
Property (d)
Abstract in English
In this thesis we work in the context of Banach spaces and Banach lattices. In a first instant, we studied the property (d) in Banach lattices and the class of almost limited completely continuous operators on Banach lattices. For example, we give a new characterization of this property in terms of the solid hull of almost limited sets and we give conditions on the Banach lattices $E$ and $F$ in order to study the majorization problem for almost limited completely continuous operators. In the following, we introduced two new classes of subsets in Banach lattices in order to localize the notions of weak Grothendieck property and positive Grothendieck property. We compared theses new classes with some quite related different classes. This allows us to introduce and compare the corresponding linear operators. In the setting of Banach lattices, it is important to obtain examples of Banach lattices that satisfy the already known geometric properties. Thus we study such properties in the Banach lattices (⨁_{n=1}^∞ \ell_2^n ight )_0, (⨁_{n=1}^∞ \ell_2^n ight )_1 e (⨁_{n=1}^∞ \ell_2^n ight )_∞. Finally, returning to the context of Banach spaces, we considered weak* $p$-summable sequences in the dual $X'$ in order to introduce and study a wide new class of subsets of a Banach space $X$ named coarse $p$-limited sets. We study its basic properties and compare it with the class of compact and weakly compact sets. Results concerning the relationship of coarse $p$-limited sets with operators are obtained.
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