On irreducibility of Gaussian quantum Markov semigroups

The generator of a Gaussian quantum Markov semigroup on the algebra of bounded operator on a Formula: see text-mode Fock space is represented in a generalized GKLS form with an operator Formula: see text quadratic in creation and annihilation operators and Kraus operators Formula: see text linear in creation and annihilation operators. Kraus operators, commutators Formula: see text and iterated commutators Formula: see text up to the order Formula: see text, as linear combinations of creation and annihilation operators determine a vector in Formula: see text. We show that a Gaussian quantum Markov semigroup is irreducible if such vectors generate Formula: see text, under the technical condition that the domains of Formula: see text 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 Formula: see text is fully non-commutative.