  The relaxation of a dissipative system to its equilibrium state often shows a
multiexponential pattern with relaxation rates, which are typically considered
to be independent of the initial condition. The rates follow from the spectrum
of a Hermitian operator obtained by a similarity transformation of the initial
Fokker-Planck operator. However, some initial conditions are mapped by this
similarity transformation to functions which grow at infinity. These cannot be
expanded in terms of the eigenfunctions of an Hermitian operator, and show
different relaxation patterns. Considering the exactly solvable examples of
Gaussian and generalized L\'evy Ornstein-Uhlenbeck processes (OUPs) we show
that the relaxation rates belong to the Hermitian spectrum only if the initial
condition belongs to the domain of attraction of the stable distribution
defining the noise. While for an ordinary OUP initial conditions leading to
non-spectral relaxation can be considered exotic, for generalized OUPs driven
by L\'evy noise these are the rule.
