Teoria de Perron Frobenius para mapas positivos

Abstract

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.