  We revisit conservative/dissipative and positive/null decompositions of
stationary max-stable processes. Originally, both decompositions were defined
in an abstract way based on the underlying non-singular flow representation. We
provide simple criteria which allow to tell whether a given spectral function
belongs to the conservative/dissipative or positive/null part of the de Haan
spectral representation. Specifically, we prove that a spectral function is
null-recurrent iff it converges to $0$ in the Ces\`{a}ro sense. For processes
with locally bounded sample paths we show that a spectral function is
dissipative iff it converges to $0$. Surprisingly, for such processes a
spectral function is integrable a.s. iff it converges to $0$ a.s. Based on
these results, we provide new criteria for ergodicity, mixing, and existence of
a mixed moving maximum representation of a stationary max-stable process in
terms of its spectral functions. In particular, we study a decomposition of
max-stable processes which characterizes the mixing property.
