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Doctoral Thesis
DOI
https://doi.org/10.11606/T.45.2020.tde-22032020-222349
Document
Author
Full name
André Luis Porto da Silva
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2019
Supervisor
Committee
Galego, Eloi Medina (President)
Aurichi, Leandro Fiorini
Bianconi, Ricardo
Lopes, Vinicius Cifú
Silva, Antonio Roberto da
Title in Portuguese
Versões não-lineares e vetoriais do teorema de Banach-Stone
Keywords in Portuguese
Constante de Schäffer
Quasi-isometrias
Teorema de Amir-Cambern
Teorema de Banach-Stone
Abstract in Portuguese
Seja X um espaço de Banach de dimensão finita e K, S espaços de Hausdorff localmente compactos. Nessa tese de doutorado, lidamos com o problema de quando uma função T de C_0(K,X) sobre C_0(S,X) implica que K e S são homeomorfos. Para esse propósito, apresentamos uma nova técnica, inspirada na prova de um resultado clássico de Jarosz (1989), que nos dá versões do teorema de Banach-Stone para funções bijetoras T: C_(K,X) \to C_(S, X) satisfazendo \frac \|f-g\|-L \leq \|T(f)-T(g)\|\leq M \|f-g\|+L, para toda f, g \in C_(K, X). Esse é o resultado de um projeto de longa data, desde o trabalho de mestrado do autor, e envolveu um extenso estudo de artigos escritos por Cambern, Jarosz, Dutriex, Kalton, Górak, entre outros. No que segue, formalizamos essa técnica, depois discutimos os resultados provenientes dela e apresentamos as provas detalhadas dos dois teoremas mais importantes. O primeiro teorema garante que K e S são homeomorfos sempre que L \geq 0 e 1 \leq M^< \mathcal S(X), onde \mathcal S(X) denota a constante de Sch\"affer de X, estendendo e unificando alguns resultados lineares e vetoriais para o contexto não-linear. O segundo teorema nos dá uma extensão da versão clássica do teorema de Banach-Stone para espaços de Hilbert, provada por Cambern, para isomorfismos com distorção maior que \sqrt, resolvendo um antigo problema em aberto.
Title in English
Nonlinear and vector versions of the Banach-Stone theorem
Keywords in English
Amir-Cambern theorem
Banach-Stone theorem
Quasi-isometries
Schäffer's constant
Abstract in English
Let X be a finite-dimensional Banach space and K, S be locally compact Hausdorff spaces. In this doctoral thesis, we deal with the problem of whether a map T from C_0(K,X) onto C_0(S,X) implies that K and S are homeomorphic. For that purpose, we present a brand new technique, inspired by the proof of a classical result by Jarosz (1989), which gives us versions of the Banach-Stone theorem for bijective maps T: C_(K,X) \to C_(S, X) satisfying \frac \|f-g\|-L \leq \|T(f)-T(g)\|\leq M \|f-g\|+L, for every f, g \in C_(K, X). This is the result of a longstanding project, since the author's masters work, and involved the extensive study of several papers writen by Cambern, Jarosz, Dutriex, Kalton, Górak, among others. In what follows, we formalize this technique, then we discuss the results provided by it and present detailed proofs of the two most important theorems. The first one states that K and S are homeomorphic whenever L \geq 0 and 1 \leq M^< \mathcal S(X), where \mathcal S(X) denotes the Sch\"affer constant of X, extending and unifying several linear versions to a nonlinear context. The second one provides an extension of the classical linear version of Banach-Stone theorem for Hilbert spaces, proved by Cambern, to isomorphisms with distortion greater than \sqrt, solving a longstanding open problem.
 
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Publishing Date
2020-04-22
 
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