A note on invariant states of Gaussian quantum Markov semigroups

Gaussian quantum Markov semigroup on the algebra of bounded operator on a d-mode Fock space is characterized either by the property of their predual that preserves quantum Gaussian states. In this paper, show that gaussian QMSs can have only Gaussian invariant states if the spectrum of the drift matrix does not contain points on the imaginary axis. Moreover, in this case, any initial state converges in trace norm toward a unique gaussian invariant state.