  Quantum trajectories are Markov processes modeling the evolution of a quantum
system subjected to repeated independent measurements. Inspired by the theory
of random products of matrices, it has been shown that these Markov processes
admit a unique invariant measure under a purification and an irreducibility
assumptions. This paper is devoted to the spectral study of the underlying
Markov operator. Using Quasi-compactness, it is shown that this operator admits
a spectral gap and the peripheral spectrum is described in a precise manner.
Next two perturbations of this operator are studied. This allows to derive
limit theorems (Central Limit Theorem, Berry-Esseen bounds and Large Deviation
Principle) for the empirical mean of functions of the Markov chain as well as
the Lyapounov exponent of the underlying random dynamical system.
