Critérios para as Componentes Espectrais Absolutamente Contínuas de Operadores de Jacobi a Valores Matriciais

Abstract

We dicuss in this work some characterizations of the absolutely continuous spectrum of arbitrary multiplicity of discrete matrix-valued Jacobi operators on (...). Essentially, these results extend some well-known characterizations for scalar Schrödinger operators. The first result obtained is an extension of the Jitomirskaya-Last inequality (Jitomirskaya & Last [1999]), which relates the asymptoptic behavior of the solutions to the eingenvalue equation to the spectral measures. Such characterizations are extensions of Kotani's theory (Kotani & Simon[1988]) for ergodic operators, and of Last-Simon results (Last & Simon [1999]), which relate the absolutely continuous spectrum to the assintoptical behavior of the Cesàro mean of the transfer matrices' norms. Especifically, we prove (as in Kotani & Simon [1988]) that for an integer (...), the absolutely continuous spectrum of multiplicity (...) is related to the energies for which the (...) smaller Lyapunov exponents are 0. We also prove that the absolutely continuous spectrum of multiplicity (...) is also related to the asymptoptic behavior, in (...), of the Cesàro mean of the singular values of some matrices given by solutions to the eigenvalue equation. This result allows us to prove the constancy of these spectral components for minimal Jacobi operators. In this setting, we give a proof of a weaker version of the exponential dichotomy (Haro & Puig [2013], Johnson [1986], Marx [2014]). Namely, we prove that the resolvent set is the set of energies for which the operator's cocicle satisfies an uniformly exponential growth condition.We also discuss Kotani's theory for ergodic Dirac operators, which can be seen as particular class of singular matrix-valued Jacobi operators.