On irreducibility of Gaussian quantum Markov semigroups

The generator of a Gaussian quantum Markov semigroup on the algebra of bounded operator on a d-mode Fock space is represented in a generalized GKLS form with an operator G quadratic in creation and annihilation operators and Kraus operators L1,... ,Lm linear in creation and annihilation operators. Kraus operators, commutators |G,L-l| and iterated commutators |G, |G,L-l||, horizontal ellipsis up to the order 2d - m, as linear combinations of creation and annihilation operators determine a vector in DOUBLE-STRUCK CAPITAL C-2d. We show that a Gaussian quantum Markov semigroup is irreducible if such vectors generate DOUBLE-STRUCK CAPITAL C-2d, under the technical condition that the domains of G and the number operator coincide. Conversely, we show that this condition is also necessary if the linear space generated by Kraus operators and their iterated commutator with G is fully non-commutative.