  We study spectral and steady-state properties of generic Markovian
dissipative systems described by quadratic fermionic Liouvillian operators of
the Lindblad form. The Hamiltonian dynamics is modeled by a generic random
quadratic operator, i.e., as a featureless superconductor of class D, whereas
the Markovian dissipation is described by $M$ random linear jump operators. By
varying the dissipation strength and the ratio of dissipative channels per
fermion, $m=M/(2N_F)$, we find two distinct phases where the support of the
single-particle spectrum has one or two connected components. In the strongly
dissipative regime, this transition occurs for $m=1/2$ and is concomitant with
a qualitative change in both the steady-state and the spectral gap that rules
the large-time dynamics. Above this threshold, the spectral gap and the
steady-state purity qualitatively agree with the fully generic (i.e.,
non-quadratic) case studied recently. Below $m=1/2$, the spectral gap closes in
the thermodynamic limit and the steady-state decouples into an ergodic and a
nonergodic sector yielding a non-monotonic steady-state purity as a function of
the dissipation strength. Our results show that some of the universal features
previously observed for fully random Liouvillians are generic for a
sufficiently large number of jump operators. On the other hand, if the number
of dissipation channels is decreased the system can exhibit nonergodic
features, rendering it possible to suppress dissipation in protected subspaces
even in the presence of strong system-environment coupling.
