We discuss the mathematical structure of quantum states on which the master
equation generates a semigroup of irreversible time evolution. As an illus
tration of an open system we choose to treat the harmonic oscillator, in wh
ich case we can perform the construction of a complete set of eigenelements
describing both the attenuator and the amplifier. We show that the corresp
onding eigenelements belong to adjoint spaces, and their orthogonality and
completeness is shown explicitly. We also show that our results are in agre
ement with earlier work by Briegel and Englert. We illustrate the use of ou
r results by deriving previously known time-dependent results for some simp
le cases.