Spectral decomposition of the Lindblad operator

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.