Spectral Theory

Abstract

Let P be a positive Markov kernel on (Formula Presented) admitting an invariant distribution (Formula Presented). We have shown that P defines an operator on the Banach space. Therefore, a natural approach to the properties of P consists in studying the spectral properties of this operator. This is the main theme of this chapter, in which we first define the spectrum of P seen as an operator both on, (Formula Presented), and on an appropriately defined space of complex measures. We will also define the adjoint operator and establish some key relations between the operator norm of the operator and that of its adjoint. We also discuss geometric and exponential convergence in (Formula Presented). We show that the existence of an (Formula Presented) -spectral gap implies (Formula Presented)-geometric ergodicity; these two notions are shown to be equivalent if the operator P is self-adjoint in (Formula Presented) (or equivalently that (Formula Presented) is reversible with respect to P). We extend these notions to cover exponential convergence in. In we introduce the notion of conductance and establish the Cheeger inequality for reversible Markov kernels.