  We point out that the quantum dynamical map of an open quantum system can be
generated by an effective Liouville operator. The effective Liouville shows the
dynamical breaking of time reversibility. This breaking of reversibility is
expressed by the effective Liouville's discrete spectrum having negative
imaginary parts. This generated dynamics of open quantum systems is capable of
memory effects described by a frequency dependence of the spectrum. When memory
effects can be neglected or smoothed out, the effective Liouville generates the
well known semi-group dynamics of open systems in the Markov approximation. The
spectral analysis of the effective Liouville -with or without memory- allows to
represent the quantum dynamical map and expectation values of physical
quantities by metric expressions using complex eigenvalues and bi-orthonormal
eigenmodes. The long time dynamics is dominated by the zero-mode, which forms
the stationary state of the open system. The relaxation time scale $\tau$ is
set by the inverse of the smallest negative imaginary part of the non-vanishing
effective eigenvalues of the effective Liouville. In generic cases of
system-environment-coupling this time scale $\tau$ is the relevant scale for
both, the relaxation of diagonal elements (populations) and the decay of
off-diagonal elements (coherences) of an initial density matrix in the
stationary state's diagonal representation. It also sets the time scale after
which the entropy tends to its stationary value. The effective Liouville and
its spectral content thus give a framework to study the dynamic breaking of
time reversibility, memory effects, decoherence, relaxation to a stationary
state and the role of entropy in open quantum systems.
