Dissipators for symmetric quasi-free dynamical semigroups on the CAR algebra

Abstract

Let L be a negative self-adjoint bounded operator on a Hilbert space H , and p a projection on H with pLp trace class, and let { T t : t ⩾ 0} be the extension of { e tL : t ⩾ 0} to a strongly continuous semigroup of completely positive quasi-free unital maps of Fock type on the fermion algebra A H built over H . Then it is shown that there exists a strongly continuous self-adjoint contraction semigroup { G t : t ⩾ 0} on the Hilbert space of the GNS decomposition of the quasi-free state gw p such that in the representation of that state: T t ⩾ G t (·) G t , t ⩾0.