We present several results and constructions from operator theory that will prove fundamental for the study of Ritt operators. This includes some basics of spectral theory, the Mean ergodic theorem on reflexive Banach spaces and the Dunford-Riesz functional calculus on Banach algebras. We also give the Katznelson-Tzafriri theorem which characterizes power bounded operators T on Banach space such that the difference \(T^n -T^{n-1}\) tends to zero.