Teoria de Perron Frobenius para mapas positivos

• Autor(a) principal: Oliveira, Leonardo Silva de
• Data de Publicação: 2019
• Tipo de documento: Trabalho de conclusão de curso
• Idioma: por
• Título da fonte: Repositório Institucional da UFU
• Texto Completo: https://repositorio.ufu.br/handle/123456789/27834
• topic: Mapas positivos; Positive maps; Raio espectral; Spectral radius; Teorema de Perron-Frobenius; Mapas irredutíveis; Mapas completamente redutíveis; Perron-Frobenius theorem; Irreducible maps; Completely reducible maps; CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA

In this work we study Perron-Frobenius theory for positive maps acting on the matrix algebra $M_k$ and its subalgebras VMkV={ VXV | X pertencendo à Mk}. We show that the spectral radius of any positive map belongs to its spectrum and associated to this eigenvalue there is a positive semidefinite hermitian eigenvector. Moreover, if the map is irreducible then we show that the geometric multiplicity of the spectral radius is 1 and the image of the associated eigenvector coincides with the image of V. We also study positive maps which are direct sum of irreducible maps and we provide an indirect way to construct them.