  In this paper we explore the set of linear maps sending the set of quantum
Gaussian states into itself. These maps are in general not positive, a feature
which can be exploited as a test to check whether a given quantum state belongs
to the convex hull of Gaussian states (if one of the considered maps sends it
into a non positive operator, the above state is certified not to belong to the
set). Generalizing a result known to be valid under the assumption of complete
positivity, we provide a characterization of these Gaussian-to-Gaussian (not
necessarily positive) superoperators in terms of their action on the
characteristic function of the inputs. For the special case of one-mode
mappings we also show that any Gaussian-to-Gaussian superoperator can be
expressed as a concatenation of a phase-space dilatation, followed by the
action of a completely positive Gaussian channel, possibly composed with a
transposition. While a similar decomposition is shown to fail in the multi-mode
scenario, we prove that it still holds at least under the further hypothesis of
homogeneous action on the covariance matrix.
