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Master's Dissertation
DOI
10.11606/D.45.2017.tde-01122017-214259
Document
Author
Full name
Jeovanny de Jesus Muentes Acevedo
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2013
Supervisor
Committee
Benevieri, Pierluigi (President)
Brech, Christina
Federson, Márcia Cristina Anderson Braz
Title in Portuguese
O fluxo espectral de caminhos de operadores de Fredholm auto-adjuntos em espaços de Hilbert
Keywords in Portuguese
Espaços de Hilbert
Fluxo espectral
Índice de Morse
Operadores de Fredholm
Teoria espectral.
Abstract in Portuguese
O objetivo principal desta dissertação é apresentar o fluxo espectral de um caminho de operadores de Fredholm auto-adjuntos em um espaço de Hilbert e suas propriedades. Pelos resultados clássicos de teoria espectral, sabemos que se H é um espaço de Hilbert e L : H → H é um operador linear, limitado e auto-adjunto, H pode ser escrito como soma direta ortogonal H+(L)⊕ H-(L)⊕ Ker L, onde H+(L) e H-(L) são os subespaços espectrais positivo e negativo de L, respectivamente. No trabalho damos uma definição de fluxo espectral baseada na decomposição acima, aprofundando as conexões deste conceito com a teoria espectral dos operadores de Fredholm em espaços de Hilbert. Entre as propriedades do fluxo espectral, será analisada a invariância homotópica que se apresenta em várias formas. Veremos o conceito de índice de Morse relativo, que estende o clássico índice de Morse, e sua relação com o fluxo espectral. A construção do fluxo espectral dada neste trabalho segue a abordagem de P. M. Fitzpatrick, J. Pejsachowicz e L. Recht em [9].
Title in English
Spectral flow of a path of selfadjoint Fredholm operators in Hilbert spaces
Keywords in English
Fredholm operators
Hilbert spaces
Morse index
Spectral flow
Spectral theory.
Abstract in English
The main purpose of this dissertation is to present the spectral flow of a path of selfadjoint Fredholm operators in a Hilbert space and its properties. By classical results in spectral theory, we know that, if H is a Hilbert space and L : H → H is a bounded self-adjoint linear operator, H may be written as the following orthogonal direct sum H = H+(L)⊕ H-(L)⊕ Ker L, where H+(L) and H-(L) are the positive and negative spectral subspaces of L, respectively. In this work we give a definition of spectral flow which is based on the above splitting, examining in depth the connection between this concept and the spectral theory of Fredholm operators in Hilbert spaces. Among the properties of the spectral flow we will analyze the homotopic invariance, which appears on different ways. We will see the concept of relative Morse index, which generalize the classical Morse index, and its relation with the spectral flow. The construction of the spectral flow given in this work follows the approach of P. M. Fitzpatrick, J. Pejsachowicz and L. Recht in [9].
 
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Publishing Date
2017-12-05
 
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